Plane Geometry — Lines, Triangles, Quadrilaterals and Circles — Extra Practice

Competitive Mathematics · Geometry

7 Part15 worked examples602 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

In geometry we look at how a figure is built (which side equals or is parallel to which) and use that to find angles and lengths. A problem may look hard, but it is usually a mix of a handful of rules — the angles of a triangle add up to $18 0^{\circ}$ , angles on parallel lines, Pythagoras, similarity and the circle theorems. This chapter takes the rules one at a time and solves new questions on each.

∑ Formula

त्रिभुज : ∠ A + ∠ B + ∠ C = 18 0^{\circ}; चतुर्भुज : 36 0^{\circ}; n - भुज : (n - 2) 18 0^{\circ}

Three angle-sum rules

One habit that helps in every question: write every angle on the figure as you find it. The next angle usually becomes visible by itself.

Lines and angles

Complementary, supplementary and parallel-line angles

The kinds of angle, angles on a line, vertically opposite angles and angles made by a transversal.

Kinds of angle and their relations

Pairs that make

9 0^{\circ}

and

18 0^{\circ}

Acute, right, obtuse, straight and reflex angles; complementary and supplementary angles; vertically opposite angles.

Name	Measure
Acute	$0^{\circ} < θ < 9 0^{\circ}$
Right	$θ = 9 0^{\circ}$
Obtuse	$9 0^{\circ} < θ < 18 0^{\circ}$
Straight	$θ = 18 0^{\circ}$
Reflex	$18 0^{\circ} < θ < 36 0^{\circ}$
Complementary pair	sum $9 0^{\circ}$
Supplementary pair	sum $18 0^{\circ}$

Angles on a straight line add up to $18 0^{\circ}$ ; angles all round a point add up to $36 0^{\circ}$ .
When two lines cross, the opposite (vertical) angles are equal; adjacent angles are supplementary.
An angle bisector cuts an angle into two equal parts. The bisectors of the two angles of a linear pair are always at $9 0^{\circ}$ to each other.
Ratio questions: divide the total (90, 180 or 360) by the sum of the parts, then multiply by the part you need.

Example 1 · Basic

An angle is 24° more than its complement. Find the angle and its complement.

Solution, step by step

1Let the angle be $θ$ ; its complement is $90 - θ$ . $θ - (90 - θ) = 24$ .
2 $2 θ = 114$ ⟹ $θ = 5 7^{\circ}$ ; complement $= 3 3^{\circ}$

Answer 57° and 33°

Example 2 · Standard

Three consecutive angles on a straight line are

(2 x - 10)^{\circ}

(x + 30)^{\circ}

and

(3 x + 10)^{\circ}

. Find the largest angle.

Solution, step by step

1 $6 x + 30 = 180$ ⟹ $x = 25$ .
2Angles: $4 0^{\circ}, 5 5^{\circ}, 8 5^{\circ}$ — sum $18 0^{\circ}$ ✓; the largest is $8 5^{\circ}$

Answer 85°

BasicTry it yourself first

What is the complement of 35°?

BasicTry it yourself first

What is the supplement of 65°?

StandardTry it yourself first

What is the complement of 72°?

StandardTry it yourself first

What is the supplement of 118°?

BasicTry it yourself first

An angle is

\frac{2}{3}

of its supplement. Find the angle.

StandardTry it yourself first

An angle is

\frac{1}{5}

of its supplement. Find the angle.

StandardTry it yourself first

An angle is

\frac{3}{2}

of its supplement. Find the angle.

AdvancedTry it yourself first

An angle is

\frac{4}{5}

of its supplement. Find the angle.

BasicTry it yourself first

An angle is

\frac{1}{2}

of its complement. Find the angle.

StandardTry it yourself first

An angle is

\frac{2}{3}

of its complement. Find the angle.

AdvancedTry it yourself first

An angle is

\frac{3}{2}

of its complement. Find the angle.

BasicTry it yourself first

Two complementary angles are in the ratio 4 : 5. Find the larger angle.

StandardTry it yourself first

Two complementary angles are in the ratio 2 : 7. Find the larger angle.

AdvancedTry it yourself first

Two complementary angles are in the ratio 7 : 11. Find the larger angle.

BasicTry it yourself first

Two supplementary angles are in the ratio 4 : 5. Find the smaller angle.

StandardTry it yourself first

Two supplementary angles are in the ratio 3 : 7. Find the smaller angle.

AdvancedTry it yourself first

Two supplementary angles are in the ratio 5 : 13. Find the smaller angle.

BasicTry it yourself first

Two supplementary angles differ by 20°. Find the larger angle.

StandardTry it yourself first

Two complementary angles differ by 40°. Find the larger angle.

StandardTry it yourself first

Two supplementary angles differ by 36°. Find the larger angle.

AdvancedTry it yourself first

Two complementary angles differ by 66°. Find the larger angle.

AdvancedTry it yourself first

Two supplementary angles differ by 24°. Find the larger angle.

StandardTry it yourself first

Three consecutive angles on one side of a straight line are

(x + 20)^{\circ}

(2 x)^{\circ}

and

(x - 20)^{\circ}

. Find

x

StandardTry it yourself first

Three consecutive angles on one side of a straight line are

(x + 10)^{\circ}

(3 x)^{\circ}

and

(2 x + 20)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(x + 30)^{\circ}

(2 x + 10)^{\circ}

and

(x - 20)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(2 x - 10)^{\circ}

(3 x + 5)^{\circ}

and

(x + 35)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(5 x - 10)^{\circ}

(3 x + 20)^{\circ}

and

(2 x)^{\circ}

. Find

x

StandardTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(3 x + 10)^{\circ}

and

(5 x - 20)^{\circ}

. Find the measure of each of these angles.

StandardTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(2 x + 30)^{\circ}

and

(4 x - 10)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(5 x - 15)^{\circ}

and

(3 x + 25)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(6 x - 30)^{\circ}

and

(4 x + 10)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(7 x + 5)^{\circ}

and

(9 x - 25)^{\circ}

. Find the measure of each of these angles.

StandardTry it yourself first

AOB is a straight line and ray OC is drawn so that

∠ A OC = 5 0^{\circ}

. OP bisects

∠ A OC

and OQ bisects

∠ COB

. Find

∠ POQ

AdvancedTry it yourself first

AOB is a straight line and ray OC is drawn so that

∠ A OC = 6 4^{\circ}

. OP bisects

∠ A OC

and OQ bisects

∠ COB

. Find

∠ POQ

Parallel lines and a transversal

One line, eight angles, only two values

Corresponding, alternate and co-interior angles, and exterior angles.

◭ Diagram

∥

CD, transversal EF

When a transversal cuts two parallel lines, all eight angles take just one of two values: some $α$ , or its supplement $18 0^{\circ} - α$ . All you need to decide is which angle gets which value.

Pair	Relation
Corresponding angles	equal
Alternate angles	equal
Vertically opposite	equal
Co-interior angles	sum $18 0^{\circ}$
Linear pair	sum $18 0^{\circ}$

In a “Z” shape alternate angles are equal; in an “F” shape corresponding angles are equal; in a “C/U” shape co-interior angles are supplementary.
Bisectors of two co-interior angles are perpendicular (their halves add up to $\frac{180}{2} = 9 0^{\circ}$ ).
For a “bent” line through a point between parallels, draw a parallel through the point and think in two parts.

Example 3 · Standard

A transversal cuts two parallel lines and the co-interior angles on one side are in the ratio 7 : 2. Find both angles.

Solution, step by step

1Sum $18 0^{\circ}$ , parts $7 + 2 = 9$ , one part $= 2 0^{\circ}$ .
2Angles: $7 \times 20 = 14 0^{\circ}$ and $2 \times 20 = 4 0^{\circ}$

Answer 140° and 40°

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 6 5^{\circ}

, find the alternate angle

∠ G H C

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 6 5^{\circ}

, find the co-interior angle

∠ G HD

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 7 2^{\circ}

, find the corresponding angle

∠ DH F

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 5 8^{\circ}

, find the vertically opposite angle

∠ A GE

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 11 0^{\circ}

, find the linear-pair angle

∠ EGB

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 12 3^{\circ}

, find the alternate angle

∠ G H C

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 13 4^{\circ}

, find the co-interior angle

∠ G HD

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 4 7^{\circ}

, find the corresponding angle

∠ DH F

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 9 8^{\circ}

, find the vertically opposite angle

∠ A GE

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 7 6^{\circ}

, find the linear-pair angle

∠ EGB

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 8 9^{\circ}

, find the co-interior angle

∠ G HD

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 3 7^{\circ}

, find the alternate angle

∠ G H C

BasicTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 2 : 3. Find the larger angle.

StandardTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 4 : 5. Find the larger angle.

StandardTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 1 : 5. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 7 : 11. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 5 : 13. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 7 : 5. Find the larger angle.

Triangles

The most-asked part

Angles, sides, congruence, similarity, the four centres and the main theorems.

Angles and sides

Sum

18 0^{\circ}

, exterior angle, side–angle link

Kinds of triangle, the angle sum, the exterior angle, the side inequality and a use of Pythagoras.

Kinds: by sides — equilateral, isosceles, scalene; by angles — acute, right, obtuse.
The angles add up to $18 0^{\circ}$ . Each angle of an equilateral triangle is $6 0^{\circ}$ ; in an isosceles triangle the angles opposite equal sides are equal.
Exterior angle = sum of the two remote interior angles (the angle made by producing a side).
The bigger angle faces the longer side; any two sides together must exceed the third.
Kind from the sides: for the longest side $z$ — $x^{2} + y^{2} > z^{2}$ acute, $= z^{2}$ right, $< z^{2}$ obtuse.

Example 4 · Standard

Side BC of triangle ABC is produced to D. The exterior angle

∠ A C D = 11 0^{\circ}

and

∠ A = 4 0^{\circ}

. Find

∠ B

and the kind of triangle.

Solution, step by step

1Exterior angle $= ∠ A + ∠ B$ ⟹ $110 = 40 + ∠ B$ ⟹ $∠ B = 7 0^{\circ}$ .
2 $∠ C = 180 - 110 = 7 0^{\circ}$ , so $∠ B = ∠ C$ : an acute isosceles triangle (AB $=$ AC)

Answer 70°; acute isosceles

BasicTry it yourself first

The angles of a triangle are in the ratio 2 : 3 : 4. Find the largest angle.

BasicTry it yourself first

The angles of a triangle are in the ratio 1 : 2 : 3. Find the largest angle.

StandardTry it yourself first

The angles of a triangle are in the ratio 3 : 4 : 5. Find the largest angle.

StandardTry it yourself first

The angles of a triangle are in the ratio 2 : 5 : 5. Find the largest angle.

AdvancedTry it yourself first

The angles of a triangle are in the ratio 5 : 7 : 6. Find the largest angle.

AdvancedTry it yourself first

The angles of a triangle are in the ratio 4 : 5 : 9. Find the largest angle.

StandardTry it yourself first

The angles of a triangle are in the ratio 1 : 2 : 3. What kind of triangle is it?

StandardTry it yourself first

The angles of a triangle are in the ratio 2 : 3 : 7. What kind of triangle is it?

StandardTry it yourself first

The angles of a triangle are in the ratio 4 : 5 : 6. What kind of triangle is it?

AdvancedTry it yourself first

The angles of a triangle are in the ratio 1 : 4 : 7. What kind of triangle is it?

AdvancedTry it yourself first

The angles of a triangle are in the ratio 1 : 1 : 2. What kind of triangle is it?

AdvancedTry it yourself first

The angles of a triangle are in the ratio 2 : 2 : 5. What kind of triangle is it?

BasicTry it yourself first

Two interior angles of a triangle are 45° and 55°. Find the exterior angle at the third vertex.

BasicTry it yourself first

Two interior angles of a triangle are 62° and 48°. Find the exterior angle at the third vertex.

StandardTry it yourself first

A side of a triangle is produced, making an exterior angle of 130°; one of the two remote interior angles is 45°. Find the other remote interior angle.

StandardTry it yourself first

A side of a triangle is produced, making an exterior angle of 115°; one of the two remote interior angles is 60°. Find the other remote interior angle.

StandardTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 2 : 3. Find the larger of them.

AdvancedTry it yourself first

A side of a triangle is produced, making an exterior angle of 140°; one of the two remote interior angles is 72°. Find the other remote interior angle.

AdvancedTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 1 : 2. Find the larger of them.

AdvancedTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 5 : 7. Find the larger of them.

BasicTry it yourself first

The sides of a triangle are 5, 12 and 13 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 6, 7 and 8 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 4, 5 and 10 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 7, 8 and 15 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 5, 6 and 10 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 9, 10 and 14 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 10, 11 and 12 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 8, 15 and 17 units. What kind of triangle is it?

BasicTry it yourself first

In triangle ABC,

∠ A = 5 0^{\circ}

∠ B = 6 0^{\circ}

and

∠ C = 7 0^{\circ}

. Which is the longest side?

StandardTry it yourself first

In triangle ABC,

∠ A = 4 0^{\circ}

∠ B = 4 5^{\circ}

and

∠ C = 9 5^{\circ}

. Which is the longest side?

StandardTry it yourself first

In triangle ABC,

∠ A = 3 0^{\circ}

∠ B = 10 0^{\circ}

and

∠ C = 5 0^{\circ}

. Which is the shortest side?

AdvancedTry it yourself first

In triangle ABC,

∠ A = 6 5^{\circ}

∠ B = 3 5^{\circ}

and

∠ C = 8 0^{\circ}

. Which is the shortest side?

The main theorems

Proportion, mid-point, median and similarity

The angle bisector theorem, basic proportionality, the mid-point theorem, Apollonius and similar triangles.

Theorem	Statement
Angle bisector	If the bisector of $∠ A$ meets BC at D then $\frac{B D}{D C} = \frac{A B}{A C}$
Basic proportionality (Thales)	If DE $∥$ BC then $\frac{A D}{D B} = \frac{A E}{EC}$ and $\frac{D E}{BC} = \frac{A D}{A B}$
Mid-point theorem	The line joining the mid-points of two sides is parallel to the third and half of it
Apollonius	For median AD, $A B^{2} + A C^{2} = 2 (A D^{2} + B D^{2})$
Altitude in a right triangle	$C D^{2} = A D \cdot D B$ and $A C^{2} = A D \cdot A B$
Similar triangles	Ratio of areas = square of the ratio of corresponding sides

Congruence (everything equal): SSS, SAS, ASA, AAS, and RHS for right triangles.
Similarity (same shape, different size): AA, SAS (sides proportional) and SSS (all three sides proportional).
In a right triangle the mid-point of the hypotenuse is equidistant from all three vertices, so the median to the hypotenuse = half the hypotenuse.

Example 5 · Standard

In triangle ABC, AB

= 10

cm, AC

= 15

cm and BC

= 20

cm. The bisector of

∠ A

meets BC at D. Find BD and DC.

Solution, step by step

1 $\frac{B D}{D C} = \frac{10}{15} = \frac{2}{3}$ .
2 $B D = \frac{2}{5} \times 20 = 8$ cm; $D C = 12$ cm

Answer 8 cm and 12 cm

Example 6 · Standard

In triangle ABC, DE

∥

BC with D on AB and E on AC. AD

= 5

cm, DB

= 10

cm, BC

= 21

cm. Find DE.

Solution, step by step

1 $A B = 5 + 10 = 15$ ; $\frac{D E}{BC} = \frac{A D}{A B} = \frac{5}{15} = \frac{1}{3}$ .
2 $D E = \frac{21}{3} = 7$ cm

Answer 7 cm

BasicTry it yourself first

In triangle ABC, AB

= 6

cm, AC

= 9

cm and BC

= 10

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, AB

= 8

cm, AC

= 12

cm and BC

= 15

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, AB

= 10

cm, AC

= 14

cm and BC

= 12

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 6

cm and BC

= 20

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 12

cm, AC

= 18

cm and BC

= 25

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 21

cm, AC

= 14

cm and BC

= 35

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

BasicTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 4

cm, DB

= 6

cm and AE

= 5

cm, find EC (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 3

cm, DB

= 5

cm and AE

= 6

cm, find EC (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 6

cm, DB

= 9

cm and AE

= 8

cm, find EC (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 5

cm, DB

= 10

cm and AE

= 7

cm, find EC (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 8

cm, DB

= 12

cm and AE

= 10

cm, find EC (cm).

BasicTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 3

cm, DB

= 2

cm and BC

= 15

cm, find DE (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 4

cm, DB

= 6

cm and BC

= 20

cm, find DE (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 5

cm, DB

= 3

cm and BC

= 16

cm, find DE (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 6

cm, DB

= 4

cm and BC

= 25

cm, find DE (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 2

cm, DB

= 3

cm and BC

= 20

cm, find DE (cm).

BasicTry it yourself first

D and E are the mid-points of sides AB and AC of triangle ABC. If BC

= 18

cm, find DE (cm).

StandardTry it yourself first

D and E are the mid-points of AB and AC of triangle ABC. If BC

-

= 7

cm, find BC (cm).

StandardTry it yourself first

A triangle has perimeter 40 cm. Find the perimeter of the triangle formed by joining the mid-points of its sides (cm).

AdvancedTry it yourself first

D and E are the mid-points of AB and AC of triangle ABC. If BC

-

= 13

cm, find BC (cm).

AdvancedTry it yourself first

A triangle has perimeter 90 cm. Find the perimeter of the triangle formed by joining the mid-points of its sides (cm).

StandardTry it yourself first

In triangle ABC, AB

= 7

cm, AC

= 9

cm and BC

= 14

cm. Find the length of the median AD on BC (cm).

StandardTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 13

cm and BC

= 20

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 11

cm, AC

= 17

cm and BC

= 26

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 10

cm, AC

= 20

cm and BC

= 26

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 19

cm and BC

= 22

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 7

cm, AC

= 9

cm and BC

= 8

cm. Find the length of the median AD on BC (cm).

BasicTry it yourself first

The sides containing the right angle of a right triangle are 6 cm and 8 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right triangle are 8 cm and 15 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right triangle are 20 cm and 21 cm. Find the length of the median on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right triangle are 9 cm and 40 cm. Find the length of the median on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right triangle are 12 cm and 35 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 2

cm and BD

= 18

cm, find CD (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 8

cm and BD

= 18

cm, find CD (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 9

cm and BD

= 16

cm, find CD (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 5

cm and BD

= 20

cm, find CD (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 4

cm and BD

= 5

cm, find the side AC (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 9

cm and BD

= 16

cm, find the side AC (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 8

cm and BD

= 10

cm, find the side AC (cm).

StandardTry it yourself first

Two similar triangles have areas 18 sq cm and 50 sq cm. A side of the first is 6 cm. Find the corresponding side of the second (cm).

StandardTry it yourself first

Two similar triangles have areas 48 sq cm and 75 sq cm. A side of the first is 8 cm. Find the corresponding side of the second (cm).

AdvancedTry it yourself first

Two similar triangles have areas 27 sq cm and 12 sq cm. A side of the first is 9 cm. Find the corresponding side of the second (cm).

AdvancedTry it yourself first

Two similar triangles have areas 45 sq cm and 20 sq cm. A side of the first is 12 cm. Find the corresponding side of the second (cm).

AdvancedTry it yourself first

Two similar triangles have areas 50 sq cm and 98 sq cm. A side of the first is 10 cm. Find the corresponding side of the second (cm).

The four centres of a triangle

Centroid, incentre, circumcentre, orthocentre

Which lines make each centre, and the angle formulas attached to them.

◭ Diagram

The three medians meet at the centroid G, which divides each median

2 : 1

from the vertex

Centre	Meeting of	Special property
Centroid	medians	divides a median $2 : 1$
Incentre	angle bisectors	centre of the incircle; $r = \frac{Δ}{s}$ ; $∠ B I C = 9 0^{\circ} + \frac{A}{2}$
Circumcentre	perpendicular bisectors	centre of the circumcircle; $R = \frac{ab c}{4Δ}$ ; $∠ BOC = 2 A$
Orthocentre	altitudes	$∠ B H C = 18 0^{\circ} - A$

In an equilateral triangle all four centres coincide.
In a right triangle: the circumcentre is the mid-point of the hypotenuse ( $R = \frac{hypotenuse}{2}$ ); the orthocentre is at the right-angled vertex; the inradius is $r = \frac{p + b - h}{2}$ .
Circumcentre: $OB = OC$ , so $∠ OBC = 9 0^{\circ} - A$ (in an acute triangle).

Example 7 · Standard

In triangle ABC,

∠ B = 7 0^{\circ}

and

∠ C = 5 0^{\circ}

; I is the incentre. Find

∠ B I C

Solution, step by step

1 $∠ A = 180 - 70 - 50 = 6 0^{\circ}$ .
2 $∠ B I C = 90 + \frac{60}{2} = 12 0^{\circ}$ (check: $\frac{70}{2} + \frac{50}{2} = 60$ , $180 - 60 = 120$ ✓)

Answer 120°

Example 8 · Standard

Find the inradius of the triangle with sides 13 cm, 14 cm and 15 cm.

Solution, step by step

1 $s = 21$ ; area $= 21 \times 8 \times 7 \times 6 = 84$ .
2 $r = \frac{84}{21} = 4$ cm

Answer 4 cm

BasicTry it yourself first

The median AD of a triangle is 18 cm and G is the centroid. Find AG (cm).

BasicTry it yourself first

The median AD of a triangle is 21 cm and G is the centroid. Find GD (cm).

StandardTry it yourself first

G is the centroid of a triangle and AG

= 14

cm on the median AD. Find the whole median AD (cm).

StandardTry it yourself first

The median AD of a triangle is 27 cm and G is the centroid. Find AG (cm).

AdvancedTry it yourself first

The median AD of a triangle is 33 cm and G is the centroid. Find GD (cm).

AdvancedTry it yourself first

G is the centroid of a triangle and AG

= 22

cm on the median AD. Find the whole median AD (cm).

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 3 5^{\circ}

and circumcentre O is given. Find

∠ OBC

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 6 4^{\circ}

and circumcentre O is given. Find

∠ OBC

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 4 2^{\circ}

and circumcentre O is given. Find

∠ OBC

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 3 8^{\circ}

and circumcentre O is given. Find

∠ BOC

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 4 7^{\circ}

and circumcentre O is given. Find

∠ BOC

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 6 4^{\circ}

and incentre I is given. Find

∠ B I C

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 8 0^{\circ}

and incentre I is given. Find

∠ B I C

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 5 2^{\circ}

and incentre I is given. Find

∠ B I C

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 5 8^{\circ}

and orthocentre H is given. Find

∠ B H C

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 7 2^{\circ}

and orthocentre H is given. Find

∠ B H C

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 4 6^{\circ}

and orthocentre H is given. Find

∠ B H C

StandardTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 6 cm, 8 cm and 10 cm (cm).

StandardTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 5 cm, 12 cm and 13 cm (cm).

StandardTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 8 cm, 15 cm and 17 cm (cm).

AdvancedTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 20 cm, 21 cm and 29 cm (cm).

AdvancedTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 13 cm, 13 cm and 24 cm (cm).

AdvancedTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 7 cm, 24 cm and 25 cm (cm).

StandardTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 6 cm, 8 cm and 10 cm (cm).

StandardTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 5 cm, 12 cm and 13 cm (cm).

AdvancedTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 8 cm, 15 cm and 17 cm (cm).

AdvancedTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 20 cm, 21 cm and 29 cm (cm).

AdvancedTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 13 cm, 13 cm and 24 cm (cm).

AdvancedTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 9 cm, 12 cm and 15 cm (cm).

Quadrilaterals and polygons

Four sides and more

Properties of quadrilaterals, the cyclic quadrilateral, mid-point figures and polygon angles.

Properties of quadrilaterals

The diagonals tell you what the figure is

Parallelogram, rectangle, rhombus, square, trapezium and cyclic quadrilateral.

Figure	Main properties
Parallelogram	opposite sides and angles equal; adjacent angles supplementary; diagonals bisect each other
Rectangle	all angles $9 0^{\circ}$ ; diagonals equal
Rhombus	four equal sides; diagonals bisect at right angles
Square	both a rectangle and a rhombus; diagonals equal and perpendicular
Trapezium	exactly one pair of parallel sides

◭ Diagram

Cyclic quadrilateral ABCD: all four vertices on the circle

The angles of a quadrilateral add up to $36 0^{\circ}$ .
Cyclic quadrilateral: opposite angles are supplementary ( $∠ A + ∠ C = 18 0^{\circ}$ ); the exterior angle made by producing a side = the interior opposite angle.
Joining the mid-points in order gives: a parallelogram from any quadrilateral (half the area); a rhombus from a rectangle; a rectangle from a rhombus; a square from a square.
A parallelogram inscribed in a circle is a rectangle; a parallelogram circumscribed about a circle is a rhombus.

Example 9 · Standard

One angle of a parallelogram is 12° less than 3 times its smallest angle. Find all four angles.

Solution, step by step

1Adjacent angles are supplementary: $x + (3 x - 12) = 180$ ⟹ $4 x = 192$ ⟹ $x = 4 8^{\circ}$ .
2Angles: $4 8^{\circ}, 13 2^{\circ}, 4 8^{\circ}, 13 2^{\circ}$ (sum $36 0^{\circ}$ ✓)

Answer 48°, 132°, 48°, 132°

BasicTry it yourself first

Three angles of a quadrilateral are 70°, 105° and 95°. Find the fourth angle.

BasicTry it yourself first

Three angles of a quadrilateral are 65°, 110° and 88°. Find the fourth angle.

StandardTry it yourself first

Three angles of a quadrilateral are 84°, 76° and 120°. Find the fourth angle.

StandardTry it yourself first

Three angles of a quadrilateral are 100°, 85° and 75°. Find the fourth angle.

StandardTry it yourself first

The angles of a quadrilateral are in the ratio 2 : 3 : 5 : 8. Find the smallest angle.

StandardTry it yourself first

The angles of a quadrilateral are in the ratio 1 : 2 : 4 : 5. Find the largest angle.

AdvancedTry it yourself first

The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 5. Find the largest angle.

AdvancedTry it yourself first

The angles of a quadrilateral are in the ratio 4 : 5 : 6 : 9. Find the smallest angle.

AdvancedTry it yourself first

The angles of a quadrilateral are in the ratio 1 : 3 : 3 : 5. Find the largest angle.

StandardTry it yourself first

One angle of a parallelogram is 20° less than 3 times its smallest angle. Find its largest angle.

StandardTry it yourself first

One angle of a parallelogram is 30° less than 2 times its smallest angle. Find its largest angle.

AdvancedTry it yourself first

One angle of a parallelogram is 10° less than 4 times its smallest angle. Find its largest angle.

AdvancedTry it yourself first

One angle of a parallelogram is 40° less than 3 times its smallest angle. Find its largest angle.

AdvancedTry it yourself first

One angle of a parallelogram is 30° less than 5 times its smallest angle. Find its largest angle.

AdvancedTry it yourself first

One angle of a parallelogram is 20° less than 4 times its smallest angle. Find its largest angle.

BasicTry it yourself first

In a cyclic quadrilateral ABCD,

∠ A = 7 8^{\circ}

. Find

∠ C

StandardTry it yourself first

In a cyclic quadrilateral ABCD,

∠ A = 10 1^{\circ}

. Find

∠ C

StandardTry it yourself first

Side AB of the cyclic quadrilateral ABCD is produced to E. If

∠ A D C = 9 5^{\circ}

, find the exterior angle

∠ CBE

AdvancedTry it yourself first

Side AB of the cyclic quadrilateral ABCD is produced to E. If

∠ A D C = 11 2^{\circ}

, find the exterior angle

∠ CBE

StandardTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 4 : 5. Find the larger one.

AdvancedTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 7 : 11. Find the larger one.

AdvancedTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 2 : 7. Find the larger one.

StandardTry it yourself first

Joining the mid-points of the four sides of a rectangle in order gives which figure?

StandardTry it yourself first

Joining the mid-points of the four sides of a rhombus in order gives which figure?

StandardTry it yourself first

Joining the mid-points of the four sides of a square in order gives which figure?

StandardTry it yourself first

Joining the mid-points of the four sides of any quadrilateral in order gives which figure?

AdvancedTry it yourself first

Joining the mid-points of the four sides of a parallelogram in order gives which figure?

StandardTry it yourself first

The mid-points of the sides of a 6 cm × 8 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

StandardTry it yourself first

The mid-points of the sides of a 5 cm × 12 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

AdvancedTry it yourself first

The mid-points of the sides of a 8 cm × 15 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

AdvancedTry it yourself first

The mid-points of the sides of a 24 cm × 7 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

AdvancedTry it yourself first

The mid-points of the sides of a 9 cm × 40 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

StandardTry it yourself first

A quadrilateral has area 84 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

StandardTry it yourself first

A quadrilateral has area 150 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

AdvancedTry it yourself first

A quadrilateral has area 96 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

AdvancedTry it yourself first

A quadrilateral has area 250 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

Polygon angles

(n - 2) \times 18 0^{\circ}

and

36 0^{\circ}

The sum of interior and exterior angles; finding each angle and the number of sides in a regular polygon.

Sum of interior angles $= (n - 2) \times 18 0^{\circ}$ ; the exterior angles always add up to $36 0^{\circ}$ .
Regular polygon: each exterior angle $= \frac{360}{n}$ ; each interior $= 180 -$ exterior.
If interior : exterior $= p : q$ , the exterior angle is $\frac{180 q}{p + q}$ , then $n = \frac{360}{exterior}$ .
If each interior exceeds the exterior by $d$ , then $180 - 2 e = d$ ⟹ $e = \frac{180 - d}{2}$ .

Example 10 · Standard

The interior angles of a polygon add up to 1260°. Find the number of sides and, if it is regular, each angle.

Solution, step by step

1 $(n - 2) \times 180 = 1260$ ⟹ $n - 2 = 7$ ⟹ $n = 9$ .
2Each angle $= \frac{1260}{9} = 14 0^{\circ}$

Answer 9 sides; 140°

BasicTry it yourself first

Find the sum of all the interior angles of a polygon with 7 sides.

BasicTry it yourself first

Find the sum of all the interior angles of a polygon with 10 sides.

StandardTry it yourself first

Find the sum of all the interior angles of a polygon with 12 sides.

StandardTry it yourself first

Find the sum of all the interior angles of a polygon with 15 sides.

AdvancedTry it yourself first

Find the sum of all the interior angles of a polygon with 20 sides.

StandardTry it yourself first

The interior angles of a polygon add up to 1080°. How many sides does it have?

StandardTry it yourself first

The interior angles of a polygon add up to 1620°. How many sides does it have?

AdvancedTry it yourself first

The interior angles of a polygon add up to 1980°. How many sides does it have?

AdvancedTry it yourself first

The interior angles of a polygon add up to 2520°. How many sides does it have?

AdvancedTry it yourself first

The interior angles of a polygon add up to 3960°. How many sides does it have?

StandardTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 90°. How many sides does it have?

StandardTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 108°. How many sides does it have?

StandardTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 120°. How many sides does it have?

AdvancedTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 132°. How many sides does it have?

AdvancedTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 140°. How many sides does it have?

AdvancedTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 100°. How many sides does it have?

StandardTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 3 : 1. How many sides does it have?

StandardTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 5 : 1. How many sides does it have?

AdvancedTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 7 : 2. How many sides does it have?

AdvancedTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 4 : 1. How many sides does it have?

AdvancedTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 13 : 2. How many sides does it have?

AdvancedTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 9 : 1. How many sides does it have?

The circle

Angles, chords and tangents

Central and inscribed angles, chord properties, tangents, secants and common tangents.

Angles and chords in a circle

The angle at the centre is double

The central angle, the angle in a semicircle, the alternate segment and a chord’s distance.

◭ Diagram

Arc AB makes

∠ A OB

at the centre and

∠ A CB = \frac{1}{2} ∠ A OB

on the circle

The angle at the centre = twice the angle at any point of the circle. All angles in the same segment are equal.
The angle in a semicircle is $9 0^{\circ}$ (standing on a diameter).
The angle between a tangent and a chord = the angle in the alternate segment.
The perpendicular from the centre bisects the chord: $(\frac{chord}{2})^{2} + d^{2} = r^{2}$ .
The line of centres is the perpendicular bisector of the common chord of two circles.

Example 11 · Standard

In a circle with centre O, arc AB subtends 126° at the centre. Find

∠ A CB

at a point C on the major part, and

∠ A D B

for a point D on the minor arc.

Solution, step by step

1 $∠ A CB = \frac{126}{2} = 6 3^{\circ}$ .
2ACBD is cyclic: $∠ A D B = 180 - 63 = 11 7^{\circ}$

Answer 63° and 117°

Example 12 · Standard

How far from the centre is a 30 cm chord of a circle of radius 17 cm?

Solution, step by step

1Half the chord $= 15$ .
2Distance $= 1 7^{2} - 1 5^{2} = 64 = 8$ cm

Answer 8 cm

BasicTry it yourself first

An arc AB of a circle subtends 80° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

StandardTry it yourself first

An arc AB of a circle subtends 110° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

StandardTry it yourself first

An arc AB of a circle subtends 130° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

AdvancedTry it yourself first

An arc AB of a circle subtends 150° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

BasicTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 2 8^{\circ}

, find

∠ CB A

StandardTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 3 5^{\circ}

, find

∠ CB A

StandardTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 4 7^{\circ}

, find

∠ CB A

AdvancedTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 6 4^{\circ}

, find

∠ CB A

StandardTry it yourself first

The angle between the tangent at A and the chord AB is 42°. What angle does the chord AB subtend at a point C in the alternate segment?

StandardTry it yourself first

The angle between the tangent at A and the chord AB is 57°. What angle does the chord AB subtend at a point C in the alternate segment?

AdvancedTry it yourself first

The angle between the tangent at A and the chord AB is 63°. What angle does the chord AB subtend at a point C in the alternate segment?

AdvancedTry it yourself first

The angle between the tangent at A and the chord AB is 71°. What angle does the chord AB subtend at a point C in the alternate segment?

BasicTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 2 5^{\circ}

, find

∠ A OB

StandardTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 3 2^{\circ}

, find

∠ A OB

StandardTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 4 0^{\circ}

, find

∠ A OB

AdvancedTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 1 8^{\circ}

, find

∠ A OB

AdvancedTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 5 5^{\circ}

, find

∠ A OB

BasicTry it yourself first

Find the length of a chord of a circle of radius 5 cm lying 3 cm from the centre (cm).

StandardTry it yourself first

Find the length of a chord of a circle of radius 13 cm lying 5 cm from the centre (cm).

StandardTry it yourself first

Find the length of a chord of a circle of radius 10 cm lying 6 cm from the centre (cm).

AdvancedTry it yourself first

Find the length of a chord of a circle of radius 25 cm lying 7 cm from the centre (cm).

AdvancedTry it yourself first

Find the length of a chord of a circle of radius 26 cm lying 10 cm from the centre (cm).

StandardTry it yourself first

How far from the centre is a chord of length 16 cm in a circle of radius 10 cm (cm)?

StandardTry it yourself first

How far from the centre is a chord of length 24 cm in a circle of radius 13 cm (cm)?

AdvancedTry it yourself first

How far from the centre is a chord of length 48 cm in a circle of radius 25 cm (cm)?

AdvancedTry it yourself first

How far from the centre is a chord of length 24 cm in a circle of radius 15 cm (cm)?

StandardTry it yourself first

A chord 30 cm long of a circle is 8 cm from the centre. Find the radius (cm).

StandardTry it yourself first

A chord 24 cm long of a circle is 9 cm from the centre. Find the radius (cm).

AdvancedTry it yourself first

A chord 40 cm long of a circle is 15 cm from the centre. Find the radius (cm).

AdvancedTry it yourself first

A chord 48 cm long of a circle is 7 cm from the centre. Find the radius (cm).

StandardTry it yourself first

Circles of radii 13 cm and 15 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 24

cm, find the distance between the centres (cm).

AdvancedTry it yourself first

Circles of radii 10 cm and 17 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 16

cm, find the distance between the centres (cm).

AdvancedTry it yourself first

Circles of radii 13 cm and 20 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 24

cm, find the distance between the centres (cm).

AdvancedTry it yourself first

Circles of radii 25 cm and 17 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 30

cm, find the distance between the centres (cm).

Tangents and secants

From an outside point to the circle

Tangent length, the tangent–secant theorem, intersecting chords, common tangents and circumscribed figures.

◭ Diagram

From an outside point P, tangent PT and secant PAB:

P T^{2} = P A \times PB

The radius is perpendicular to the tangent at the point of contact, so $P T^{2} = O P^{2} - r^{2}$ .
Two tangents from one point are equal; if the angle between them is $θ$ , the chord of contact subtends $18 0^{\circ} - θ$ at the centre.
Tangent–secant: $P T^{2} = P A \times PB$ (take the whole secant $PB$ ). Intersecting chords: $A P \times PB = CP \times P D$ .
Common tangents: direct $= d^{2} - (R - r)^{2}$ ; transverse $= d^{2} - (R + r)^{2}$ .
Circumscribed quadrilateral: $A B + C D = BC + D A$ . Incircle of a triangle: the tangent length from vertex B is $s - b$ ( $b$ the opposite side).

Example 13 · Standard

A tangent is drawn from a point P 17 cm from the centre O of a circle of radius 8 cm. Find the length of the tangent.

Solution, step by step

1 $OT ⊥ PT$ ⟹ $P T^{2} = 1 7^{2} - 8^{2} = 289 - 64 = 225$ .
2 $PT = 15$ cm

Answer 15 cm

Example 14 · Standard

Two circles of radii 14 cm and 2 cm have centres 20 cm apart. Find the length of the direct common tangent.

Solution, step by step

1 $R - r = 12$ .
2Length $= 2 0^{2} - 1 2^{2} = 256 = 16$ cm

Answer 16 cm

BasicTry it yourself first

A tangent is drawn to a circle from a point P 13 cm from the centre. If the radius is 5 cm, find the length of the tangent (cm).

StandardTry it yourself first

A tangent is drawn to a circle from a point P 25 cm from the centre. If the radius is 7 cm, find the length of the tangent (cm).

StandardTry it yourself first

A tangent is drawn to a circle from a point P 10 cm from the centre. If the radius is 6 cm, find the length of the tangent (cm).

AdvancedTry it yourself first

A tangent is drawn to a circle from a point P 26 cm from the centre. If the radius is 10 cm, find the length of the tangent (cm).

AdvancedTry it yourself first

A tangent is drawn to a circle from a point P 37 cm from the centre. If the radius is 12 cm, find the length of the tangent (cm).

StandardTry it yourself first

The tangent PT from an external point P to a circle is 12 cm long and P is 13 cm from the centre. Find the radius of the circle (cm).

AdvancedTry it yourself first

The tangent PT from an external point P to a circle is 24 cm long and P is 25 cm from the centre. Find the radius of the circle (cm).

AdvancedTry it yourself first

The tangent PT from an external point P to a circle is 24 cm long and P is 26 cm from the centre. Find the radius of the circle (cm).

BasicTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 4

cm and PB

= 9

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 3

cm and PB

= 12

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 2

cm and PB

= 18

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 5

cm and PB

= 20

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 8

cm and PB

= 18

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 9

cm and PB

= 16

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 6

cm and PB

= 24

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 7

cm and PB

= 28

cm (A comes first). Find the length of the tangent PT from P (cm).

BasicTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 6

cm, PB

= 8

cm and CP

= 4

cm, find PD (cm).

StandardTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 9

cm, PB

= 4

cm and CP

= 6

cm, find PD (cm).

StandardTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 10

cm, PB

= 6

cm and CP

= 12

cm, find PD (cm).

AdvancedTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 8

cm, PB

= 9

cm and CP

= 6

cm, find PD (cm).

AdvancedTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 12

cm, PB

= 5

cm and CP

= 15

cm, find PD (cm).

AdvancedTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 14

cm, PB

= 6

cm and CP

= 7

cm, find PD (cm).

StandardTry it yourself first

Two circles of radii 8 cm and 3 cm have their centres 13 cm apart. Find the length of their direct common tangent (cm).

StandardTry it yourself first

Two circles of radii 12 cm and 4 cm have their centres 17 cm apart. Find the length of their direct common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 10 cm and 3 cm have their centres 25 cm apart. Find the length of their direct common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 18 cm and 8 cm have their centres 26 cm apart. Find the length of their direct common tangent (cm).

StandardTry it yourself first

Two circles of radii 3 cm and 2 cm have their centres 13 cm apart. Find the length of their transverse common tangent (cm).

StandardTry it yourself first

Two circles of radii 7 cm and 3 cm have their centres 26 cm apart. Find the length of their transverse common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 8 cm and 4 cm have their centres 37 cm apart. Find the length of their transverse common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 5 cm and 3 cm have their centres 17 cm apart. Find the length of their transverse common tangent (cm).

StandardTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 12

cm, BC

= 9

cm and CD

= 10

cm, find DA (cm).

StandardTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 15

cm, BC

= 11

cm and CD

= 13

cm, find DA (cm).

AdvancedTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 18

cm, BC

= 14

cm and CD

= 16

cm, find DA (cm).

AdvancedTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 25

cm, BC

= 18

cm and CD

= 16

cm, find DA (cm).

StandardTry it yourself first

In triangle ABC, BC

= 13

cm, CA

= 14

cm and AB

= 15

cm. The incircle touches BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, BC

= 10

cm, CA

= 12

cm and AB

= 14

cm. The incircle touches BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, BC

= 9

cm, CA

= 13

cm and AB

= 16

cm. The incircle touches BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, BC

= 11

cm, CA

= 15

cm and AB

= 18

cm. The incircle touches BC at D. Find BD (cm).

StandardTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 4 0^{\circ}

. Find

∠ A OB

StandardTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 5 6^{\circ}

. Find

∠ A OB

AdvancedTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 7 0^{\circ}

. Find

∠ A OB

AdvancedTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 8 4^{\circ}

. Find

∠ A OB

Traps and fast tricks

Slips · tricks

Where marks go in geometry questions.

The commonest mistakes

Seven traps

The slips that cost marks.

Trap 1Mixing up complementary and supplementary. Complementary adds to

9 0^{\circ}

, supplementary to

18 0^{\circ}

Trap 2Treating alternate and co-interior angles alike. Alternate (Z) angles are equal; co-interior (C) angles add up to

18 0^{\circ}

Trap 3Putting the sides the wrong way round in the angle bisector theorem.

\frac{B D}{D C} = \frac{A B}{A C}

— BD goes with AB, the side on its own side.

Trap 4Adding the exterior angle to the adjacent interior one. Exterior = sum of the two remote interior angles; with the adjacent one it makes

18 0^{\circ}

Trap 5Using only the outer part of a secant. In

P T^{2} = P A \times PB

PB

is the whole secant, not

A B

Trap 6Confusing direct and transverse common tangents. Direct uses

R - r

, transverse uses

R + r

(and is the shorter one).

Trap 7Using $n$ instead of $n - 2$ in a polygon. Interior angles add up to

(n - 2) \times 18 0^{\circ}

, not

n \times 18 0^{\circ}

Fast tricks

Recognise and apply

Habits for speed.

Fill in angles as you go: write each angle on the figure; the next usually follows from “straight line 180”, “triangle 180” or “parallel”.
Angle formulas at a glance: incentre $90 + \frac{A}{2}$ , orthocentre $180 - A$ , circumcentre $2 A$ , exterior angle = sum of two remote.
The Pythagorean triples ( $3 - 4 - 5, 5 - 12 - 13, 8 - 15 - 17, 7 - 24 - 25$ ) keep appearing in chord, tangent and median questions.
If you see a parallel line, think similar triangles or proportion; if you see a circle, recall “angle at the centre is double” and “angle in a semicircle is $9 0^{\circ}$ ”.
Check: add the angles back (triangle 180, quadrilateral 360) and test the side inequality.

Example 15 · Standard

In a cyclic quadrilateral ABCD,

∠ A = 3 x

∠ B = 2 x + 10

and

∠ C = 3 x + 30

. Find

x

and

∠ D

Solution, step by step

1Opposite angles are supplementary: $∠ A + ∠ C = 180$ ⟹ $6 x + 30 = 180$ ⟹ $x = 25$ .
2 $∠ B = 2 \times 25 + 10 = 6 0^{\circ}$ , so $∠ D = 180 - 60 = 12 0^{\circ}$ (check: $75 + 60 + 105 + 120 = 360$ ✓)

Answer $x = 25$ ; $∠ D = 12 0^{\circ}$

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

BasicTry it yourself first

What is the complement of 18°?

BasicTry it yourself first

What is the supplement of 144°?

BasicTry it yourself first

An angle is

\frac{1}{2}

of its supplement. Find the angle.

BasicTry it yourself first

An angle is

\frac{1}{5}

of its complement. Find the angle.

BasicTry it yourself first

Two complementary angles are in the ratio 1 : 2. Find the larger angle.

BasicTry it yourself first

Two supplementary angles are in the ratio 2 : 3. Find the smaller angle.

BasicTry it yourself first

Two complementary angles differ by 30°. Find the larger angle.

BasicTry it yourself first

Three consecutive angles on one side of a straight line are

(x)^{\circ}

(2 x)^{\circ}

and

(3 x)^{\circ}

. Find

x

BasicTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(2 x + 15)^{\circ}

and

(3 x + 5)^{\circ}

. Find the measure of each of these angles.

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 8 2^{\circ}

, find the corresponding angle

∠ DH F

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 3 7^{\circ}

, find the vertically opposite angle

∠ A GE

BasicTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 1 : 2. Find the larger angle.

BasicTry it yourself first

AOB is a straight line and ray OC is drawn so that

∠ A OC = 7 2^{\circ}

. OP bisects

∠ A OC

and OQ bisects

∠ COB

. Find

∠ POQ

BasicTry it yourself first

The angles of a triangle are in the ratio 1 : 1 : 2. Find the largest angle.

BasicTry it yourself first

The angles of a triangle are in the ratio 1 : 1 : 1. What kind of triangle is it?

BasicTry it yourself first

Two interior angles of a triangle are 38° and 67°. Find the exterior angle at the third vertex.

BasicTry it yourself first

The sides of a triangle are 3, 4 and 5 units. What kind of triangle is it?

BasicTry it yourself first

The sides of a triangle are 6, 6 and 6 units. What kind of triangle is it?

BasicTry it yourself first

In triangle ABC,

∠ A = 8 0^{\circ}

∠ B = 4 5^{\circ}

and

∠ C = 5 5^{\circ}

. Which is the longest side?

BasicTry it yourself first

In triangle ABC, AB

= 4

cm, AC

= 6

cm and BC

= 15

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

BasicTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 2

cm, DB

= 4

cm and AE

= 3

cm, find EC (cm).

BasicTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 1

cm, DB

= 2

cm and BC

= 12

cm, find DE (cm).

BasicTry it yourself first

D and E are the mid-points of sides AB and AC of triangle ABC. If BC

= 26

cm, find DE (cm).

BasicTry it yourself first

The sides containing the right angle of a right triangle are 3 cm and 4 cm. Find the length of the median on the hypotenuse (cm).

BasicTry it yourself first

The median AD of a triangle is 12 cm and G is the centroid. Find AG (cm).

BasicTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 1

cm and BD

= 16

cm, find CD (cm).

BasicTry it yourself first

Two similar triangles have areas 4 sq cm and 9 sq cm. A side of the first is 6 cm. Find the corresponding side of the second (cm).

BasicTry it yourself first

In an acute-angled triangle ABC,

∠ A = 3 0^{\circ}

and circumcentre O is given. Find

∠ OBC

BasicTry it yourself first

In an acute-angled triangle ABC,

∠ A = 4 0^{\circ}

and incentre I is given. Find

∠ B I C

BasicTry it yourself first

In an acute-angled triangle ABC,

∠ A = 6 0^{\circ}

and orthocentre H is given. Find

∠ B H C

BasicTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 9 cm, 12 cm and 15 cm (cm).

BasicTry it yourself first

Three angles of a quadrilateral are 90°, 90° and 80°. Find the fourth angle.

BasicTry it yourself first

The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find the smallest angle.

BasicTry it yourself first

One angle of a parallelogram is 12° less than 2 times its smallest angle. Find its largest angle.

BasicTry it yourself first

In a cyclic quadrilateral ABCD,

∠ A = 6 6^{\circ}

. Find

∠ C

BasicTry it yourself first

Joining the mid-points of the four sides of a square in order gives which figure?

BasicTry it yourself first

The mid-points of the sides of a 3 cm × 4 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

BasicTry it yourself first

A quadrilateral has area 60 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

BasicTry it yourself first

Find the sum of all the interior angles of a polygon with 6 sides.

BasicTry it yourself first

The interior angles of a polygon add up to 540°. How many sides does it have?

BasicTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 144°. How many sides does it have?

BasicTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 2 : 1. How many sides does it have?

BasicTry it yourself first

An arc AB of a circle subtends 60° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

BasicTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 3 0^{\circ}

, find

∠ CB A

BasicTry it yourself first

The angle between the tangent at A and the chord AB is 50°. What angle does the chord AB subtend at a point C in the alternate segment?

BasicTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 2 0^{\circ}

, find

∠ A OB

BasicTry it yourself first

Find the length of a chord of a circle of radius 13 cm lying 12 cm from the centre (cm).

BasicTry it yourself first

How far from the centre is a chord of length 8 cm in a circle of radius 5 cm (cm)?

BasicTry it yourself first

A chord 6 cm long of a circle is 4 cm from the centre. Find the radius (cm).

BasicTry it yourself first

A tangent is drawn to a circle from a point P 5 cm from the centre. If the radius is 3 cm, find the length of the tangent (cm).

BasicTry it yourself first

The tangent PT from an external point P to a circle is 4 cm long and P is 5 cm from the centre. Find the radius of the circle (cm).

BasicTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 1

cm and PB

= 4

cm (A comes first). Find the length of the tangent PT from P (cm).

BasicTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 4

cm, PB

= 6

cm and CP

= 3

cm, find PD (cm).

BasicTry it yourself first

Two circles of radii 4 cm and 1 cm have their centres 5 cm apart. Find the length of their direct common tangent (cm).

BasicTry it yourself first

Two circles of radii 2 cm and 1 cm have their centres 5 cm apart. Find the length of their transverse common tangent (cm).

BasicTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 8

cm, BC

= 5

cm and CD

= 7

cm, find DA (cm).

BasicTry it yourself first

In triangle ABC, BC

= 5

cm, CA

= 5

cm and AB

= 6

cm. The incircle touches BC at D. Find BD (cm).

BasicTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 6 0^{\circ}

. Find

∠ A OB

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

What is the complement of 49°?

StandardTry it yourself first

What is the supplement of 127°?

StandardTry it yourself first

An angle is

\frac{5}{4}

of its supplement. Find the angle.

StandardTry it yourself first

An angle is

\frac{4}{5}

of its complement. Find the angle.

StandardTry it yourself first

Two complementary angles are in the ratio 5 : 13. Find the larger angle.

StandardTry it yourself first

Two supplementary angles are in the ratio 4 : 11. Find the smaller angle.

StandardTry it yourself first

Two supplementary angles differ by 50°. Find the larger angle.

StandardTry it yourself first

Two complementary angles differ by 14°. Find the larger angle.

StandardTry it yourself first

Three consecutive angles on one side of a straight line are

(2 x)^{\circ}

(3 x + 10)^{\circ}

and

(x + 20)^{\circ}

. Find

x

StandardTry it yourself first

Three consecutive angles on one side of a straight line are

(2 x + 15)^{\circ}

(2 x + 5)^{\circ}

and

(2 x - 20)^{\circ}

. Find

x

StandardTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(4 x - 5)^{\circ}

and

(2 x + 35)^{\circ}

. Find the measure of each of these angles.

StandardTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(3 x + 22)^{\circ}

and

(5 x - 8)^{\circ}

. Find the measure of each of these angles.

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 7 8^{\circ}

, find the alternate angle

∠ G H C

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 10 3^{\circ}

, find the co-interior angle

∠ G HD

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 6 4^{\circ}

, find the corresponding angle

∠ DH F

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 11 5^{\circ}

, find the vertically opposite angle

∠ A GE

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 4 9^{\circ}

, find the linear-pair angle

∠ EGB

StandardTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 3 : 7. Find the larger angle.

StandardTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 2 : 13. Find the larger angle.

StandardTry it yourself first

AOB is a straight line and ray OC is drawn so that

∠ A OC = 3 6^{\circ}

. OP bisects

∠ A OC

and OQ bisects

∠ COB

. Find

∠ POQ

StandardTry it yourself first

The angles of a triangle are in the ratio 3 : 5 : 10. Find the largest angle.

StandardTry it yourself first

The angles of a triangle are in the ratio 2 : 3 : 5. Find the largest angle.

StandardTry it yourself first

The angles of a triangle are in the ratio 3 : 4 : 2. What kind of triangle is it?

StandardTry it yourself first

The angles of a triangle are in the ratio 1 : 5 : 6. What kind of triangle is it?

StandardTry it yourself first

The angles of a triangle are in the ratio 1 : 2 : 6. What kind of triangle is it?

StandardTry it yourself first

The angles of a triangle are in the ratio 3 : 3 : 4. What kind of triangle is it?

StandardTry it yourself first

Two interior angles of a triangle are 53° and 71°. Find the exterior angle at the third vertex.

StandardTry it yourself first

A side of a triangle is produced, making an exterior angle of 125°; one of the two remote interior angles is 38°. Find the other remote interior angle.

StandardTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 1 : 5. Find the larger of them.

StandardTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 7 : 3. Find the larger of them.

StandardTry it yourself first

The sides of a triangle are 8, 10 and 12 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 9, 12 and 15 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 6, 8 and 11 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 2, 3 and 6 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 13, 14 and 15 units. What kind of triangle is it?

StandardTry it yourself first

The sides of a triangle are 11, 12 and 23 units. What kind of triangle is it?

StandardTry it yourself first

In triangle ABC,

∠ A = 7 0^{\circ}

∠ B = 5 5^{\circ}

and

∠ C = 5 5^{\circ}

. Which is the longest side?

StandardTry it yourself first

In triangle ABC,

∠ A = 3 5^{\circ}

∠ B = 8 5^{\circ}

and

∠ C = 6 0^{\circ}

. Which is the shortest side?

StandardTry it yourself first

In triangle ABC,

∠ A = 4 7^{\circ}

∠ B = 7 3^{\circ}

and

∠ C = 6 0^{\circ}

. Which is the longest side?

StandardTry it yourself first

In triangle ABC, AB

= 14

cm, AC

= 21

cm and BC

= 30

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, AB

= 18

cm, AC

= 12

cm and BC

= 25

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, AB

= 20

cm, AC

= 30

cm and BC

= 36

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 15

cm and BC

= 24

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 2

cm, DB

= 5

cm and AE

= 3

cm, find EC (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 7

cm, DB

= 14

cm and AE

= 9

cm, find EC (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 9

cm, DB

= 12

cm and AE

= 6

cm, find EC (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 7

cm, DB

= 5

cm and BC

= 36

cm, find DE (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 3

cm, DB

= 9

cm and BC

= 20

cm, find DE (cm).

StandardTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 2

cm, DB

= 5

cm and BC

= 28

cm, find DE (cm).

StandardTry it yourself first

D and E are the mid-points of sides AB and AC of triangle ABC. If BC

= 34

cm, find DE (cm).

StandardTry it yourself first

D and E are the mid-points of AB and AC of triangle ABC. If BC

-

= 11

cm, find BC (cm).

StandardTry it yourself first

A triangle has perimeter 66 cm. Find the perimeter of the triangle formed by joining the mid-points of its sides (cm).

StandardTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 17

cm and BC

= 16

cm. Find the length of the median AD on BC (cm).

StandardTry it yourself first

In triangle ABC, AB

= 11

cm, AC

= 13

cm and BC

= 16

cm. Find the length of the median AD on BC (cm).

StandardTry it yourself first

In triangle ABC, AB

= 11

cm, AC

= 13

cm and BC

= 18

cm. Find the length of the median AD on BC (cm).

StandardTry it yourself first

In triangle ABC, AB

= 12

cm, AC

= 14

cm and BC

= 14

cm. Find the length of the median AD on BC (cm).

StandardTry it yourself first

In triangle ABC, AB

= 8

cm, AC

= 14

cm and BC

= 18

cm. Find the length of the median AD on BC (cm).

StandardTry it yourself first

The sides containing the right angle of a right triangle are 7 cm and 24 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right triangle are 28 cm and 45 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right triangle are 33 cm and 56 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right triangle are 10 cm and 24 cm. Find the length of the median on the hypotenuse (cm).

StandardTry it yourself first

The median AD of a triangle is 30 cm and G is the centroid. Find AG (cm).

StandardTry it yourself first

The median AD of a triangle is 24 cm and G is the centroid. Find GD (cm).

StandardTry it yourself first

G is the centroid of a triangle and AG

= 16

cm on the median AD. Find the whole median AD (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 3

cm and BD

= 12

cm, find CD (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 18

cm and BD

= 32

cm, find CD (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 2

cm and BD

= 16

cm, find the side AC (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 1

cm and BD

= 8

cm, find the side AC (cm).

StandardTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 4

cm and BD

= 12

cm, find the side AC (cm).

StandardTry it yourself first

Two similar triangles have areas 12 sq cm and 27 sq cm. A side of the first is 8 cm. Find the corresponding side of the second (cm).

StandardTry it yourself first

Two similar triangles have areas 16 sq cm and 49 sq cm. A side of the first is 8 cm. Find the corresponding side of the second (cm).

StandardTry it yourself first

Two similar triangles have areas 75 sq cm and 12 sq cm. A side of the first is 10 cm. Find the corresponding side of the second (cm).

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 2 8^{\circ}

and circumcentre O is given. Find

∠ OBC

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 7 2^{\circ}

and circumcentre O is given. Find

∠ OBC

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 5 3^{\circ}

and circumcentre O is given. Find

∠ BOC

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 3 6^{\circ}

and incentre I is given. Find

∠ B I C

StandardTry it yourself first

In an acute-angled triangle ABC,

∠ A = 6 5^{\circ}

and orthocentre H is given. Find

∠ B H C

StandardTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 12 cm, 16 cm and 20 cm (cm).

StandardTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 10 cm, 24 cm and 26 cm (cm).

StandardTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 15 cm, 20 cm and 25 cm (cm).

StandardTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 12 cm, 16 cm and 20 cm (cm).

StandardTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 10 cm, 24 cm and 26 cm (cm).

StandardTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 15 cm, 20 cm and 25 cm (cm).

StandardTry it yourself first

Three angles of a quadrilateral are 77°, 103° and 88°. Find the fourth angle.

StandardTry it yourself first

The angles of a quadrilateral are in the ratio 2 : 4 : 5 : 9. Find the smallest angle.

StandardTry it yourself first

The angles of a quadrilateral are in the ratio 1 : 4 : 5 : 6. Find the largest angle.

StandardTry it yourself first

One angle of a parallelogram is 18° less than 2 times its smallest angle. Find its largest angle.

StandardTry it yourself first

One angle of a parallelogram is 24° less than 3 times its smallest angle. Find its largest angle.

StandardTry it yourself first

In a cyclic quadrilateral ABCD,

∠ A = 8 9^{\circ}

. Find

∠ C

StandardTry it yourself first

Side AB of the cyclic quadrilateral ABCD is produced to E. If

∠ A D C = 8 4^{\circ}

, find the exterior angle

∠ CBE

StandardTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 3 : 2. Find the larger one.

StandardTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 1 : 2. Find the larger one.

StandardTry it yourself first

Joining the mid-points of the four sides of a parallelogram in order gives which figure?

StandardTry it yourself first

The mid-points of the sides of a 20 cm × 21 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

StandardTry it yourself first

The mid-points of the sides of a 12 cm × 35 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

StandardTry it yourself first

A quadrilateral has area 118 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

StandardTry it yourself first

Find the sum of all the interior angles of a polygon with 14 sides.

StandardTry it yourself first

The interior angles of a polygon add up to 1800°. How many sides does it have?

StandardTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 36°. How many sides does it have?

StandardTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 8 : 1. How many sides does it have?

StandardTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 2 : 1. How many sides does it have?

StandardTry it yourself first

An arc AB of a circle subtends 100° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

StandardTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 5 3^{\circ}

, find

∠ CB A

StandardTry it yourself first

The angle between the tangent at A and the chord AB is 38°. What angle does the chord AB subtend at a point C in the alternate segment?

StandardTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 2 8^{\circ}

, find

∠ A OB

StandardTry it yourself first

Find the length of a chord of a circle of radius 17 cm lying 15 cm from the centre (cm).

StandardTry it yourself first

Find the length of a chord of a circle of radius 25 cm lying 24 cm from the centre (cm).

StandardTry it yourself first

How far from the centre is a chord of length 10 cm in a circle of radius 13 cm (cm)?

StandardTry it yourself first

How far from the centre is a chord of length 16 cm in a circle of radius 17 cm (cm)?

StandardTry it yourself first

A chord 14 cm long of a circle is 24 cm from the centre. Find the radius (cm).

StandardTry it yourself first

A chord 16 cm long of a circle is 6 cm from the centre. Find the radius (cm).

StandardTry it yourself first

A tangent is drawn to a circle from a point P 17 cm from the centre. If the radius is 15 cm, find the length of the tangent (cm).

StandardTry it yourself first

A tangent is drawn to a circle from a point P 41 cm from the centre. If the radius is 9 cm, find the length of the tangent (cm).

StandardTry it yourself first

A tangent is drawn to a circle from a point P 29 cm from the centre. If the radius is 20 cm, find the length of the tangent (cm).

StandardTry it yourself first

The tangent PT from an external point P to a circle is 20 cm long and P is 29 cm from the centre. Find the radius of the circle (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 4

cm and PB

= 25

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 10

cm and PB

= 40

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 3

cm and PB

= 27

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 2

cm and PB

= 32

cm (A comes first). Find the length of the tangent PT from P (cm).

StandardTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 5

cm, PB

= 12

cm and CP

= 6

cm, find PD (cm).

StandardTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 9

cm, PB

= 10

cm and CP

= 15

cm, find PD (cm).

StandardTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 8

cm, PB

= 15

cm and CP

= 10

cm, find PD (cm).

StandardTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 6

cm, PB

= 14

cm and CP

= 12

cm, find PD (cm).

StandardTry it yourself first

Two circles of radii 11 cm and 2 cm have their centres 15 cm apart. Find the length of their direct common tangent (cm).

StandardTry it yourself first

Two circles of radii 15 cm and 3 cm have their centres 20 cm apart. Find the length of their direct common tangent (cm).

StandardTry it yourself first

Two circles of radii 4 cm and 3 cm have their centres 25 cm apart. Find the length of their transverse common tangent (cm).

StandardTry it yourself first

Two circles of radii 4 cm and 2 cm have their centres 10 cm apart. Find the length of their transverse common tangent (cm).

StandardTry it yourself first

Two circles of radii 12 cm and 8 cm have their centres 29 cm apart. Find the length of their transverse common tangent (cm).

StandardTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 14

cm, BC

= 10

cm and CD

= 12

cm, find DA (cm).

StandardTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 22

cm, BC

= 17

cm and CD

= 19

cm, find DA (cm).

StandardTry it yourself first

In triangle ABC, BC

= 15

cm, CA

= 16

cm and AB

= 17

cm. The incircle touches BC at D. Find BD (cm).

StandardTry it yourself first

In triangle ABC, BC

= 12

cm, CA

= 13

cm and AB

= 5

cm. The incircle touches BC at D. Find BD (cm).

StandardTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 5 0^{\circ}

. Find

∠ A OB

StandardTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 6 6^{\circ}

. Find

∠ A OB

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

An angle is

\frac{7}{5}

of its supplement. Find the angle.

AdvancedTry it yourself first

An angle is

\frac{5}{4}

of its complement. Find the angle.

AdvancedTry it yourself first

Two complementary angles are in the ratio 13 : 5. Find the larger angle.

AdvancedTry it yourself first

Two supplementary angles are in the ratio 7 : 11. Find the smaller angle.

AdvancedTry it yourself first

Two complementary angles differ by 38°. Find the larger angle.

AdvancedTry it yourself first

Two supplementary angles differ by 74°. Find the larger angle.

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(4 x - 20)^{\circ}

(2 x + 10)^{\circ}

and

(3 x + 10)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(3 x + 15)^{\circ}

(x + 25)^{\circ}

and

(2 x - 10)^{\circ}

. Find

x

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(8 x - 35)^{\circ}

and

(6 x + 5)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(9 x + 10)^{\circ}

and

(7 x + 40)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 13 2^{\circ}

, find the alternate angle

∠ G H C

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 6 1^{\circ}

, find the co-interior angle

∠ G HD

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 10 8^{\circ}

, find the corresponding angle

∠ DH F

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 13 4^{\circ}

, find the linear-pair angle

∠ EGB

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 4 : 11. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 7 : 8. Find the larger angle.

AdvancedTry it yourself first

The angles of a triangle are in the ratio 4 : 7 : 9. Find the largest angle.

AdvancedTry it yourself first

The angles of a triangle are in the ratio 3 : 4 : 8. Find the largest angle.

AdvancedTry it yourself first

The angles of a triangle are in the ratio 2 : 5 : 11. What kind of triangle is it?

AdvancedTry it yourself first

The angles of a triangle are in the ratio 3 : 8 : 9. What kind of triangle is it?

AdvancedTry it yourself first

The angles of a triangle are in the ratio 4 : 5 : 3. What kind of triangle is it?

AdvancedTry it yourself first

Two interior angles of a triangle are 87° and 64°. Find the exterior angle at the third vertex.

AdvancedTry it yourself first

A side of a triangle is produced, making an exterior angle of 152°; one of the two remote interior angles is 79°. Find the other remote interior angle.

AdvancedTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 3 : 7. Find the larger of them.

AdvancedTry it yourself first

The exterior angle of a triangle is 120° and the two remote interior angles are in the ratio 1 : 9. Find the larger of them.

AdvancedTry it yourself first

The sides of a triangle are 10, 24 and 26 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 14, 15 and 21 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 12, 13 and 25 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 16, 30 and 34 units. What kind of triangle is it?

AdvancedTry it yourself first

The sides of a triangle are 17, 18 and 19 units. What kind of triangle is it?

AdvancedTry it yourself first

In triangle ABC,

∠ A = 3 3^{\circ}

∠ B = 6 7^{\circ}

and

∠ C = 8 0^{\circ}

. Which is the longest side?

AdvancedTry it yourself first

In triangle ABC,

∠ A = 5 8^{\circ}

∠ B = 2 4^{\circ}

and

∠ C = 9 8^{\circ}

. Which is the shortest side?

AdvancedTry it yourself first

In triangle ABC, AB

= 15

cm, AC

= 25

cm and BC

= 48

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 24

cm, AC

= 16

cm and BC

= 40

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 30

cm, AC

= 42

cm and BC

= 36

cm. The bisector of

∠ A

meets BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 11

cm, DB

= 22

cm and AE

= 15

cm, find EC (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 12

cm, DB

= 18

cm and AE

= 14

cm, find EC (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 9

cm, DB

= 6

cm and BC

= 40

cm, find DE (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 5

cm, DB

= 7

cm and BC

= 60

cm, find DE (cm).

AdvancedTry it yourself first

In triangle ABC, DE

∥

BC with D on AB and E on AC. If AD

= 8

cm, DB

= 4

cm and BC

= 33

cm, find DE (cm).

AdvancedTry it yourself first

D and E are the mid-points of sides AB and AC of triangle ABC. If BC

= 58

cm, find DE (cm).

AdvancedTry it yourself first

D and E are the mid-points of AB and AC of triangle ABC. If BC

-

= 19

cm, find BC (cm).

AdvancedTry it yourself first

A triangle has perimeter 142 cm. Find the perimeter of the triangle formed by joining the mid-points of its sides (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 13

cm and BC

= 20

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 9

cm, AC

= 19

cm and BC

= 20

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 11

cm, AC

= 17

cm and BC

= 12

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 11

cm, AC

= 23

cm and BC

= 30

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

In triangle ABC, AB

= 11

cm, AC

= 27

cm and BC

= 26

cm. Find the length of the median AD on BC (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right triangle are 16 cm and 63 cm. Find the length of the median on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right triangle are 36 cm and 77 cm. Find the length of the median on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right triangle are 13 cm and 84 cm. Find the length of the median on the hypotenuse (cm).

AdvancedTry it yourself first

The median AD of a triangle is 45 cm and G is the centroid. Find AG (cm).

AdvancedTry it yourself first

The median AD of a triangle is 39 cm and G is the centroid. Find GD (cm).

AdvancedTry it yourself first

G is the centroid of a triangle and AG

= 34

cm on the median AD. Find the whole median AD (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 25

cm and BD

= 36

cm, find CD (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 12

cm and BD

= 27

cm, find CD (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 16

cm and BD

= 20

cm, find the side AC (cm).

AdvancedTry it yourself first

In triangle ABC,

∠ C = 9 0^{\circ}

and CD is the perpendicular from C to the hypotenuse AB. If AD

= 25

cm and BD

= 11

cm, find the side AC (cm).

AdvancedTry it yourself first

Two similar triangles have areas 18 sq cm and 98 sq cm. A side of the first is 9 cm. Find the corresponding side of the second (cm).

AdvancedTry it yourself first

Two similar triangles have areas 128 sq cm and 50 sq cm. A side of the first is 16 cm. Find the corresponding side of the second (cm).

AdvancedTry it yourself first

Two similar triangles have areas 72 sq cm and 162 sq cm. A side of the first is 12 cm. Find the corresponding side of the second (cm).

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 3 3^{\circ}

and circumcentre O is given. Find

∠ OBC

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 6 1^{\circ}

and circumcentre O is given. Find

∠ BOC

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 7 4^{\circ}

and incentre I is given. Find

∠ B I C

AdvancedTry it yourself first

In an acute-angled triangle ABC,

∠ A = 7 9^{\circ}

and orthocentre H is given. Find

∠ B H C

AdvancedTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 13 cm, 14 cm and 15 cm (cm).

AdvancedTry it yourself first

Find the inradius (radius of the incircle) of a triangle with sides 25 cm, 25 cm and 14 cm (cm).

AdvancedTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 7 cm, 24 cm and 25 cm (cm).

AdvancedTry it yourself first

Find the circumradius (radius of the circumcircle) of a triangle with sides 9 cm, 40 cm and 41 cm (cm).

AdvancedTry it yourself first

Three angles of a quadrilateral are 118°, 96° and 67°. Find the fourth angle.

AdvancedTry it yourself first

The angles of a quadrilateral are in the ratio 5 : 6 : 8 : 11. Find the smallest angle.

AdvancedTry it yourself first

The angles of a quadrilateral are in the ratio 3 : 4 : 6 : 11. Find the largest angle.

AdvancedTry it yourself first

One angle of a parallelogram is 18° less than 5 times its smallest angle. Find its largest angle.

AdvancedTry it yourself first

One angle of a parallelogram is 36° less than 3 times its smallest angle. Find its largest angle.

AdvancedTry it yourself first

In a cyclic quadrilateral ABCD,

∠ A = 10 7^{\circ}

. Find

∠ C

AdvancedTry it yourself first

Side AB of the cyclic quadrilateral ABCD is produced to E. If

∠ A D C = 12 8^{\circ}

, find the exterior angle

∠ CBE

AdvancedTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 11 : 7. Find the larger one.

AdvancedTry it yourself first

Two opposite angles of a cyclic quadrilateral are in the ratio 3 : 7. Find the larger one.

AdvancedTry it yourself first

The mid-points of the sides of a 28 cm × 45 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

AdvancedTry it yourself first

The mid-points of the sides of a 33 cm × 56 cm rectangle are joined in order. Find the perimeter of the figure formed (cm).

AdvancedTry it yourself first

A quadrilateral has area 372 sq cm. Find the area of the parallelogram formed by joining the mid-points of its sides in order (sq cm).

AdvancedTry it yourself first

Find the sum of all the interior angles of a polygon with 25 sides.

AdvancedTry it yourself first

The interior angles of a polygon add up to 5040°. How many sides does it have?

AdvancedTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 150°. How many sides does it have?

AdvancedTry it yourself first

Each interior angle of a regular polygon exceeds each exterior angle by 160°. How many sides does it have?

AdvancedTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 11 : 1. How many sides does it have?

AdvancedTry it yourself first

The interior and exterior angles of a regular polygon are in the ratio 5 : 1. How many sides does it have?

AdvancedTry it yourself first

An arc AB of a circle subtends 170° at the centre. What angle does it subtend at a point C on the remaining part of the circle?

AdvancedTry it yourself first

AB is a diameter of a circle and C is a point on the circle. If

∠ C A B = 5 9^{\circ}

, find

∠ CB A

AdvancedTry it yourself first

The angle between the tangent at A and the chord AB is 33°. What angle does the chord AB subtend at a point C in the alternate segment?

AdvancedTry it yourself first

O is the centre of a circle and AB is a chord. If

∠ O A B = 3 4^{\circ}

, find

∠ A OB

AdvancedTry it yourself first

Find the length of a chord of a circle of radius 29 cm lying 21 cm from the centre (cm).

AdvancedTry it yourself first

Find the length of a chord of a circle of radius 37 cm lying 35 cm from the centre (cm).

AdvancedTry it yourself first

How far from the centre is a chord of length 40 cm in a circle of radius 29 cm (cm)?

AdvancedTry it yourself first

How far from the centre is a chord of length 24 cm in a circle of radius 37 cm (cm)?

AdvancedTry it yourself first

A chord 40 cm long of a circle is 21 cm from the centre. Find the radius (cm).

AdvancedTry it yourself first

A chord 18 cm long of a circle is 40 cm from the centre. Find the radius (cm).

AdvancedTry it yourself first

Circles of radii 17 cm and 10 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 16

cm, find the distance between the centres (cm).

AdvancedTry it yourself first

Circles of radii 20 cm and 15 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 24

cm, find the distance between the centres (cm).

AdvancedTry it yourself first

Circles of radii 15 cm and 13 cm intersect at A and B with their centres on opposite sides of the common chord AB. If AB

= 24

cm, find the distance between the centres (cm).

AdvancedTry it yourself first

A tangent is drawn to a circle from a point P 53 cm from the centre. If the radius is 28 cm, find the length of the tangent (cm).

AdvancedTry it yourself first

The tangent PT from an external point P to a circle is 28 cm long and P is 53 cm from the centre. Find the radius of the circle (cm).

AdvancedTry it yourself first

A tangent is drawn to a circle from a point P 65 cm from the centre. If the radius is 16 cm, find the length of the tangent (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 12

cm and PB

= 27

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 5

cm and PB

= 45

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

From an external point P a secant meets the circle at A and B, with PA

= 18

cm and PB

= 32

cm (A comes first). Find the length of the tangent PT from P (cm).

AdvancedTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 18

cm, PB

= 10

cm and CP

= 15

cm, find PD (cm).

AdvancedTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 12

cm, PB

= 21

cm and CP

= 14

cm, find PD (cm).

AdvancedTry it yourself first

Two chords AB and CD of a circle meet inside the circle at P. If AP

= 16

cm, PB

= 9

cm and CP

= 12

cm, find PD (cm).

AdvancedTry it yourself first

Two circles of radii 24 cm and 6 cm have their centres 30 cm apart. Find the length of their direct common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 17 cm and 8 cm have their centres 41 cm apart. Find the length of their direct common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 6 cm and 3 cm have their centres 41 cm apart. Find the length of their transverse common tangent (cm).

AdvancedTry it yourself first

Two circles of radii 10 cm and 4 cm have their centres 50 cm apart. Find the length of their transverse common tangent (cm).

AdvancedTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 31

cm, BC

= 24

cm and CD

= 28

cm, find DA (cm).

AdvancedTry it yourself first

Quadrilateral ABCD is circumscribed about a circle (the circle touches all four sides). If AB

= 40

cm, BC

= 33

cm and CD

= 35

cm, find DA (cm).

AdvancedTry it yourself first

In triangle ABC, BC

= 20

cm, CA

= 21

cm and AB

= 29

cm. The incircle touches BC at D. Find BD (cm).

AdvancedTry it yourself first

In triangle ABC, BC

= 17

cm, CA

= 25

cm and AB

= 26

cm. The incircle touches BC at D. Find BD (cm).

AdvancedTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 3 8^{\circ}

. Find

∠ A OB

AdvancedTry it yourself first

Two tangents PA and PB are drawn from an external point P to a circle with centre O, and

∠ A PB = 9 2^{\circ}

. Find

∠ A OB

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.