E-Math notes · Chapter 11

Congruence and Similarity: O-Level E-Math Notes

Congruent figures are exact copies; similar figures are scaled copies. By the end of this chapter you can use the tests for each, find unknown lengths with a scale factor, enlarge and reduce figures, work with scale drawings and maps, construct bisectors, and use the area and volume ratios of similar figures.

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Chapter
11

Figure 1 · SSS

9 outcomes · 28 worked examples · every step shown ✦

Congruent figures

Congruent triangles: write the vertices in matching order.

Similar figures

Similar triangles: equal angles, and sides in the same ratio.

Enlargement and reduction

Enlargement: lengths × k, angles unchanged.

By the end of this chapter

Recognise congruent and similar figures, and use the tests for congruent triangles and for similar triangles.
Use the properties of similar triangles and polygons: equal corresponding angles and proportional corresponding sides.
Enlarge and reduce a plane figure, and work with scale drawings and maps.
Construct perpendicular bisectors and angle bisectors, and use their properties.
Use the ratio of areas and the ratio of volumes of similar figures and solids.
Solve simple problems and give proofs using similarity and congruence.

Start here

Maths In Real Life

Three situations where a scale factor links a small thing to a big one. The pictures are made separately and the figures are drawn on top by script, so every number matches its problem.

$A small architect's model of a sports hall next to a drawing, with the scale and the model's measurements marked over the picture$

Maths In Real Life 1

A model of a sports hall

Architects test a design with a small model. Lengths, areas and volumes of the real building follow from the model by different powers of the scale.

A school plans a new sports hall.

A model of a sports hall is made at scale 1 : 100. The model is 12 cm high, its roof has area 150 cm² and its volume is 1500 cm³. Find the real height, roof area and volume.

Show full solutionHide solution

\displaystyle 12 \times 100 = 1200 \text{ cm} = 12 \text{ m}

length × n, then cm to m

\displaystyle 150 \times 100^2 = 1500000 \text{ cm}^2 = 150 \text{ m}^2

area × n², then ÷ 10000

\displaystyle 1500 \times 100^3 = 1500000000 \text{ cm}^3 = 1500 \text{ m}^3

volume × n³, then ÷ 1000000

Final answer

height 12 m, roof area 150 m², volume 1500 m³

$Two aquariums of the same shape, one small and one large, on a shop shelf, with the lengths and capacities marked over the picture$

Maths In Real Life 2

Two similar fish tanks

Doubling every length of a tank makes it hold eight times as much water, but it needs only four times as much glass.

A shop sells two tanks of the same shape.

Two fish tanks are similar. The lengths are 60 cm and 90 cm. The smaller holds 54 litres and needs 9600 cm² of glass. Find the capacity and the glass area of the larger tank.

Show full solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

\displaystyle k = 90 \div 60 = \frac{3}{2}

scale factor of the lengths

\displaystyle 54 \times k^3 = 54 \times \frac{27}{8} = 182.25

capacity scales by k³

\displaystyle 9600 \times k^2 = 9600 \times \frac{9}{4} = 21600

glass area scales by k²

Final answer

capacity 182.25 litres, glass area 21600 cm²

$A child and a lamp post on a path casting long shadows on the ground, with the heights and shadow lengths marked over the picture$

Maths In Real Life 3

The shadow of a lamp post

On level ground in sunlight, an upright object and its shadow form a right-angled triangle, and all such triangles at the same time of day are similar.

A child stands next to a lamp post.

A lamp post casts a shadow 12 m long when a 1.2 m tall child casts a shadow 1.8 m long beside it. How tall is the lamp post?

Show full solutionHide solution

The child and the post stand upright, and the sun's rays are parallel, so the two triangles are similar.

\displaystyle \dfrac{x}{1.2} = \dfrac{12}{1.8}

corresponding sides of similar triangles are in the same ratio

\displaystyle x = 1.2 \times 12 \div 1.8 = 8

Final answer

8 m

Supports G2.1 · G2.6.a · G2.9

Congruent figures and the congruence tests

Two figures are congruent when they have exactly the same shape and size: one fits exactly onto the other, even if it is turned over or turned round. Matching parts are called corresponding parts, and corresponding sides and angles are equal.

The Core Idea

Congruent triangles: write the vertices in matching order.

If triangles ABC and PQR are congruent, written △ABC ≡ △PQR, then A matches P, B matches Q and C matches R. So AB = PQ, BC = QR, CA = RP, and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

Case table 1: the tests for congruent triangles (marks show the equal parts)
Test	What must be equal	Picture
SSS	all three pairs of sides
SAS	two pairs of sides and the angle between them
ASA	two pairs of angles and the side between them
AAS	two pairs of angles and a side that is not between them
RHS	a right angle, the hypotenuse and one other side

Two things that are not tests: AAA (three equal angles give the same shape, but not necessarily the same size) and SSA (two sides and an angle that is not between them). The picture below shows two different triangles with the same SSA marks: AB = 8 cm, BC = 5 cm and angle A = 30°.

same marked sides and angle, but different triangles

SGSchoolKaki Tip

When you give a reason, name the test in brackets, for example "(SAS)", and list the equal parts in matching order.

SGSchoolKaki Tip

Check that the equal angle is BETWEEN the two equal sides before you write SAS. If it is not between them, you do not have a test.

Think About It

Two triangles each have angles of 40°, 60° and 80°. Must they be congruent?

Show the worked answerHide the worked answer

Take a small triangle and an enlarged copy: both have angles 40°, 60° and 80°.

try an enlargement

They have the same angles but different sizes.

so they cannot be congruent

Final answerNo. Equal angles only make the triangles similar (same shape), not congruent.

Diagram for Example 11.1

Example 11.1

Congruence statement and corresponding parts

The two triangles are congruent. The sides with the same number of ticks are equal. Write down the congruence statement, find the lengths of the three sides of triangle QRP, and find angle R.

(i)the congruence statement
(ii)the sides of triangle QRP
(iii)angle R

Show solutionHide solution

Colour keyexpand, rewrite, substitute or solve

(i)

AB has one tick, and so has QR; BC has two ticks, and so has RP; CA has three ticks, and so has PQ.

match the ticks

A is the vertex where AB and CA meet; Q is where QR and PQ meet. So A matches Q, B matches R and C matches P.

write the vertices in matching order

\displaystyle \triangle \text{ABC} \equiv \triangle \text{QRP}

SSS: all three pairs of sides are equal

(ii)

\displaystyle QR = AB = 3, \quad RP = BC = 5, \quad PQ = CA = 4

corresponding sides of congruent triangles

(iii)

\displaystyle \angle R = \angle \text{B}

corresponding angles of congruent triangles

\displaystyle \angle\text{B} = 53.1

found with the cosine rule (or by measuring)

Final answer

(i)

\triangle

ABC

\equiv

\triangle

QRP (SSS)

(ii)QR = 3 cm, RP = 5 cm, PQ = 4 cm

(iii)

53.1

° (to one decimal place)

Your Turn 11.1

The two triangles are congruent. The sides with the same number of ticks are equal. Write down the congruence statement, find the lengths of the three sides of triangle RPQ, and find angle P.

(i)the congruence statement
(ii)the sides of triangle RPQ
(iii)angle P

Diagram for Your Turn 11.1

Show worked solutionHide worked solution

Colour keyexpand, rewrite, substitute or solve

(i)

AB has one tick, and so has RP; BC has two ticks, and so has PQ; CA has three ticks, and so has QR.

match the ticks

A is the vertex where AB and CA meet; R is where RP and QR meet. So A matches R, B matches P and C matches Q.

write the vertices in matching order

\displaystyle \triangle \text{ABC} \equiv \triangle \text{RPQ}

SSS: all three pairs of sides are equal

(ii)

\displaystyle RP = AB = 5, \quad PQ = BC = 13, \quad QR = CA = 12

corresponding sides of congruent triangles

(iii)

\displaystyle \angle P = \angle \text{B}

corresponding angles of congruent triangles

\displaystyle \angle\text{B} = 67.4

found with the cosine rule (or by measuring)

Final answer

(i)

\triangle

ABC

\equiv

\triangle

RPQ (SSS)

(ii)RP = 5 cm, PQ = 13 cm, QR = 12 cm

(iii)

67.4

° (to one decimal place)

Example 11.2

Which test?

For each pair of triangles ABC and PQR, say whether the information is enough to prove that they are congruent. If it is, name the test.

(i)In triangles ABC and PQR: AB = PQ = 6 cm; BC = QR = 7 cm; angle B = angle Q = 50°.
(ii)In triangles ABC and PQR: BC = QR = 5 cm; angle A = angle P = 40°; angle B = angle Q = 60°.
(iii)In triangles ABC and PQR: AB = PQ = 5 cm; BC = QR = 8 cm; angle A = angle P = 30°.
(iv)In triangles ABC and PQR: AC = PR = 10 cm; AB = PQ = 6 cm; angle B is a right angle in both.

Show solutionHide solution

(i)

Equal parts: AB = PQ = 6 cm; BC = QR = 7 cm; angle B = angle Q = 50°.

list what is equal

These give: two sides and the angle between them are equal (SAS).

match the pattern

(ii)

Equal parts: BC = QR = 5 cm; angle A = angle P = 40°; angle B = angle Q = 60°.

list what is equal

These give: two angles and a side that is not between them are equal (AAS).

match the pattern

(iii)

Equal parts: AB = PQ = 5 cm; BC = QR = 8 cm; angle A = angle P = 30°.

list what is equal

These give: not enough: two sides and an angle that is NOT between them (SSA) does not make a congruence test.

match the pattern

(iv)

Equal parts: AC = PR = 10 cm; AB = PQ = 6 cm; angle B is a right angle in both.

list what is equal

These give: a right angle, the hypotenuse and one other side are equal (RHS).

match the pattern

Final answer

(i)congruent (SAS)

(ii)congruent (AAS)

(iii)not enough to prove congruence (SSA is not a test)

(iv)congruent (RHS)

Your Turn 11.2

For each pair of triangles ABC and PQR, say whether the information is enough to prove that they are congruent. If it is, name the test.

(i)In triangles ABC and PQR: AB = PQ = 8 cm; angle A = angle P = 35°; angle B = angle Q = 70°.
(ii)In triangles ABC and PQR: AB = PQ = 4 cm; BC = QR = 5 cm; CA = RP = 6 cm.
(iii)In triangles ABC and PQR: angle A = angle P = 40°; angle B = angle Q = 60°; angle C = angle R = 80°.
(iv)In triangles ABC and PQR: AB = PQ = 7 cm; CA = RP = 9 cm; angle A = angle P = 45°.

Show worked solutionHide worked solution

(i)

Equal parts: AB = PQ = 8 cm; angle A = angle P = 35°; angle B = angle Q = 70°.

list what is equal

These give: two angles and the side between them are equal (ASA).

match the pattern

(ii)

Equal parts: AB = PQ = 4 cm; BC = QR = 5 cm; CA = RP = 6 cm.

list what is equal

These give: all three sides are equal (SSS).

match the pattern

(iii)

Equal parts: angle A = angle P = 40°; angle B = angle Q = 60°; angle C = angle R = 80°.

list what is equal

These give: not enough for congruence: three equal angles give similar triangles, not necessarily congruent ones.

match the pattern

(iv)

Equal parts: AB = PQ = 7 cm; CA = RP = 9 cm; angle A = angle P = 45°.

list what is equal

These give: two sides and the angle between them are equal (SAS).

match the pattern

Final answer

(i)congruent (ASA)

(ii)congruent (SSS)

(iii)not enough to prove congruence (AAA gives similar triangles)

(iv)congruent (SAS)

Diagram for Example 11.3

Example 11.3

Using congruence to find unknowns

Triangles ABC and DEF are congruent. Find $x$ and $y$ .

(i)Find $x$ .
(ii)Find $y$ .

Show solutionHide solution

(i)

\displaystyle 2x + 3 = 13

AB = DE: corresponding sides of congruent triangles

\displaystyle 2x = 13 - 3 = 10

move the number across

\displaystyle x = 10 \div 2 = 5

(ii)

\displaystyle 3y - 4 = 56

angle ACB = angle DFE: corresponding angles of congruent triangles

\displaystyle 3y = 56 + 4 = 60

move the number across

\displaystyle y = 60 \div 3 = 20

Final answer

(i)

x = 5

(ii)

y = 20

Diagram for Example 11.4

Example 11.4

Proof: a kite

ABCD is a kite with AB = AD and CB = CD. (i) Prove that triangles ABC and ADC are congruent. (ii) Find angle DAC. (iii) Find angle ADC.

(i)Prove that triangles ABC and ADC are congruent.
(ii)angle DAC
(iii)angle ADC

Show solutionHide solution

(i)

In triangle ABC and triangle ADC:

AB = AD

given

CB = CD

given

AC is a side of both triangles

common side

So triangles ABC and ADC are congruent.

SSS: three pairs of equal sides

(ii)

\displaystyle \angle\text{DAC} = \angle\text{BAC} = 35

corresponding angles of congruent triangles

(iii)

\displaystyle \angle\text{ADC} = \angle\text{ABC} = 95

corresponding angles of congruent triangles

Final answer

(i)Triangles ABC and ADC are congruent (SSS).

(ii)

35^{\circ}

(iii)

95^{\circ}

Your Turn 11.4

ABCD is a kite with AB = AD and CB = CD. (i) Prove that triangles ABC and ADC are congruent. (ii) Find angle DAC. (iii) Find angle ADC.

(i)Prove that triangles ABC and ADC are congruent.
(ii)angle DAC
(iii)angle ADC

Diagram for Your Turn 11.4

Show worked solutionHide worked solution

(i)

In triangle ABC and triangle ADC:

AB = AD

given

CB = CD

given

AC is a side of both triangles

common side

So triangles ABC and ADC are congruent.

SSS: three pairs of equal sides

(ii)

\displaystyle \angle\text{DAC} = \angle\text{BAC} = 30

corresponding angles of congruent triangles

(iii)

\displaystyle \angle\text{ADC} = \angle\text{ABC} = 110

corresponding angles of congruent triangles

Final answer

(i)Triangles ABC and ADC are congruent (SSS).

(ii)

30^{\circ}

(iii)

110^{\circ}

Diagram for Example 11.5

Example 11.5

Proof: a parallelogram

ABCD is a parallelogram with AB = 7 cm and angle DAB = 64°. (i) Prove that triangles ABC and CDA are congruent. (ii) Find CD. (iii) Find angle BCD.

(i)Prove that triangles ABC and CDA are congruent.
(ii)CD
(iii)angle BCD

Show solutionHide solution

(i)

In triangle ABC and triangle CDA:

AB = CD

opposite sides of a parallelogram

BC = DA

opposite sides of a parallelogram

AC = CA

common side

So triangles ABC and CDA are congruent.

SSS

(ii)

\displaystyle CD = AB = 7

opposite sides of a parallelogram (or corresponding sides of congruent triangles)

(iii)

\displaystyle \angle\text{BCD} = \angle\text{DAB} = 64

opposite angles of a parallelogram

Final answer

(i)Triangles ABC and CDA are congruent (SSS).

(ii)7 cm

(iii)

64^{\circ}

Your Turn 11.5

ABCD is a parallelogram with AB = 6 cm and angle DAB = 110°. (i) Prove that triangles ABC and CDA are congruent. (ii) Find CD. (iii) Find angle BCD.

(i)Prove that triangles ABC and CDA are congruent.
(ii)CD
(iii)angle BCD

Diagram for Your Turn 11.5

Show worked solutionHide worked solution

(i)

In triangle ABC and triangle CDA:

AB = CD

opposite sides of a parallelogram

BC = DA

opposite sides of a parallelogram

AC = CA

common side

So triangles ABC and CDA are congruent.

SSS

(ii)

\displaystyle CD = AB = 6

opposite sides of a parallelogram (or corresponding sides of congruent triangles)

(iii)

\displaystyle \angle\text{BCD} = \angle\text{DAB} = 110

opposite angles of a parallelogram

Final answer

(i)Triangles ABC and CDA are congruent (SSS).

(ii)6 cm

(iii)

110^{\circ}

Links To

The free overview of geometry and congruence covers the same ideas in short: free E-Math revision page: Geometry & Congruence

Supports G2.1 · G2.2.a · G2.2.b · G2.6.b · G2.9

Similar figures and the similarity tests

Two figures are similar when they have the same shape but not necessarily the same size. One is an enlargement (or reduction) of the other. In similar figures the corresponding angles are equal and the corresponding sides are in the same ratio (the scale factor).

The Core Idea

Similar triangles: equal angles, and sides in the same ratio.

If △ABC is similar to △PQR then PQ/AB = QR/BC = RP/CA. This common ratio is the scale factor. Write the vertices in matching order, as with congruence.

Case table 2: the tests for similar triangles
Test	What must be true	Picture
AA (also called AAA)	two pairs of equal angles (the third pair is then equal too)
SSS	all three pairs of sides are in the same ratio
SAS	two pairs of sides are in the same ratio, and the angles between them are equal

SGSchoolKaki Tip

To find a missing side, write the equal ratios with the unknown on top, for example PQ/AB = QR/BC, then solve. Always match the sides by their vertices, not by how they look.

SGSchoolKaki Tip

For SSS similarity the ratios must be equal for ALL three pairs. Two equal ratios and a third that differs does not make the triangles similar.

Think About It

Are all squares similar to each other? Are all rectangles?

Show the worked answerHide the worked answer

Colour keyfinal answer

Two squares: every angle is a right angle and the ratio of each pair of sides is the same.

check angles, then ratios

Two rectangles: the angles are always right angles but the length-to-width ratios can differ.

so the ratios may not match

Final answerAll squares are similar. Rectangles are similar only if their length-to-width ratios are equal.

Diagram for Example 11.6

Example 11.6

Finding unknown sides in similar triangles

The triangles are similar, with the equal angles marked. Find $x$ and $y$ .

(i)the scale factor
(ii) $x$ (QR)
(iii) $y$ (RP)

Show solutionHide solution

(i)

The equal angles show that P matches A, Q matches B and R matches C.

match the vertices first

\displaystyle \text{scale factor} = \dfrac{PQ}{AB} = \dfrac{4.5}{3} = \frac{3}{2}

a pair of corresponding sides you know

(ii)

\displaystyle \dfrac{QR}{BC} = \frac{3}{2}

corresponding sides of similar triangles are in the same ratio

\displaystyle x = \frac{3}{2} \times 5 = 7.5

(iii)

\displaystyle \dfrac{RP}{CA} = \frac{3}{2}

corresponding sides of similar triangles are in the same ratio

\displaystyle y = \frac{3}{2} \times 4 = 6

Final answer

(i)

\frac{3}{2}

(ii)

x = 7.5

(iii)

y = 6

Your Turn 11.6

The triangles are similar, with the equal angles marked. Find $x$ and $y$ .

(i)the scale factor
(ii) $x$ (QR)
(iii) $y$ (RP)

Diagram for Your Turn 11.6

Show worked solutionHide worked solution

(i)

The equal angles show that P matches A, Q matches B and R matches C.

match the vertices first

\displaystyle \text{scale factor} = \dfrac{PQ}{AB} = \dfrac{4}{6} = \frac{2}{3}

a pair of corresponding sides you know

(ii)

\displaystyle \dfrac{QR}{BC} = \frac{2}{3}

corresponding sides of similar triangles are in the same ratio

\displaystyle x = \frac{2}{3} \times 13 \approx 8.67

(iii)

\displaystyle \dfrac{RP}{CA} = \frac{2}{3}

corresponding sides of similar triangles are in the same ratio

\displaystyle y = \frac{2}{3} \times 12 = 8

Final answer

(i)

\frac{2}{3}

(ii)

x = 8.67

(iii)

y = 8

Example 11.7

Which similarity test?

Decide whether each pair of triangles is similar, and name the test you use.

(i)Triangle ABC has angles 40° and 75°; triangle PQR has angles 40° and 75°.
(ii)Triangle ABC has sides 3 cm, 4 cm, 5 cm; triangle PQR has sides 6 cm, 8 cm, 10 cm.
(iii)Triangle ABC has two sides 4 cm and 6 cm with the angle 50° between them; triangle PQR has two sides 6 cm and 9 cm with the angle 50° between them.
(iv)Triangle ABC has two sides 4 cm and 6 cm with the angle 50° between them; triangle PQR has two sides 6 cm and 10 cm with the angle 50° between them.

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplifyround, estimate or compare

(i)

Triangle ABC has angles 40° and 75°; triangle PQR has angles 40° and 75°.

list the angles

The third angles are 65° and 65°: all three angles match.

angle sum of a triangle

(ii)

Triangle ABC has sides 3 cm, 4 cm, 5 cm; triangle PQR has sides 6 cm, 8 cm, 10 cm.

list the sides in order

\displaystyle \dfrac{6}{3} = 2 , \ \dfrac{8}{4} = 2 , \ \dfrac{10}{5} = 2

divide each side of PQR by the matching side of ABC

(iii)

Triangle ABC has two sides 4 cm and 6 cm with the angle 50° between them; triangle PQR has two sides 6 cm and 9 cm with the angle 50° between them.

the angle is between the two sides

\displaystyle \dfrac{6}{4} = \frac{3}{2}, \quad \dfrac{9}{6} = \frac{3}{2}

compare the two ratios

(iv)

Triangle ABC has two sides 4 cm and 6 cm with the angle 50° between them; triangle PQR has two sides 6 cm and 10 cm with the angle 50° between them.

the angle is between the two sides

\displaystyle \dfrac{6}{4} = \frac{3}{2}, \quad \dfrac{10}{6} = \frac{5}{3}

compare the two ratios

Final answer

(i)similar (AA: two equal angles are enough)

(ii)similar (SSS: all three ratios are equal)

(iii)similar (SAS: two ratios are equal and the angles between are equal)

(iv)not similar (the two ratios are different)

Your Turn 11.7

Decide whether each pair of triangles is similar, and name the test you use.

(i)Triangle ABC has angles 35° and 80°; triangle PQR has angles 35° and 70°.
(ii)Triangle ABC has sides 4 cm, 5 cm, 6 cm; triangle PQR has sides 6 cm, 7.5 cm, 9 cm.
(iii)Triangle ABC has two sides 3 cm and 5 cm with the angle 70° between them; triangle PQR has two sides 6 cm and 10 cm with the angle 70° between them.
(iv)Triangle ABC has sides 6 cm, 8 cm, 9 cm; triangle PQR has sides 9 cm, 12 cm, 15 cm.

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplifyround, estimate or compare

(i)

Triangle ABC has angles 35° and 80°; triangle PQR has angles 35° and 70°.

list the angles

The third angles are 65° and 75°: the third angles are different.

angle sum of a triangle

(ii)

Triangle ABC has sides 4 cm, 5 cm, 6 cm; triangle PQR has sides 6 cm, 7.5 cm, 9 cm.

list the sides in order

\displaystyle \dfrac{6}{4} = \frac{3}{2} , \ \dfrac{7.5}{5} = \frac{3}{2} , \ \dfrac{9}{6} = \frac{3}{2}

divide each side of PQR by the matching side of ABC

(iii)

Triangle ABC has two sides 3 cm and 5 cm with the angle 70° between them; triangle PQR has two sides 6 cm and 10 cm with the angle 70° between them.

the angle is between the two sides

\displaystyle \dfrac{6}{3} = 2, \quad \dfrac{10}{5} = 2

compare the two ratios

(iv)

Triangle ABC has sides 6 cm, 8 cm, 9 cm; triangle PQR has sides 9 cm, 12 cm, 15 cm.

list the sides in order

\displaystyle \dfrac{9}{6} = \frac{3}{2} , \ \dfrac{12}{8} = \frac{3}{2} , \ \dfrac{15}{9} = \frac{5}{3}

divide each side of PQR by the matching side of ABC

Final answer

(i)not similar (the angles are not all equal)

(ii)similar (SSS: all three ratios are equal)

(iii)similar (SAS: two ratios are equal and the angles between are equal)

(iv)not similar (the ratios are not all equal)

Diagram for Example 11.8

Example 11.8

Overlapping similar triangles

DE is parallel to BC. Find $x$ (the length BC) and the length AC.

(i)Explain why triangle ADE is similar to triangle ABC.
(ii)BC
(iii)AC

Show solutionHide solution

(i)

Angle A is common to both triangles.

shared angle

Angle ADE = angle ABC.

corresponding angles, DE // BC

Angle AED = angle ACB.

corresponding angles, DE // BC

So triangle ADE is similar to triangle ABC.

AA: equal angles

(ii)

\displaystyle AB = 6 + 3 = 9

AD + DB

\displaystyle \dfrac{DE}{BC} = \dfrac{AD}{AB} \ \Rightarrow \ \dfrac{8}{x} = \dfrac{6}{9}

corresponding sides of similar triangles are in the same ratio

\displaystyle x = 8 \times 9 \div 6 = 12

(iii)

\displaystyle \dfrac{AE}{AC} = \dfrac{AD}{AB} \ \Rightarrow \ \dfrac{5}{AC} = \dfrac{6}{9}

corresponding sides of similar triangles are in the same ratio

\displaystyle AC = 5 \times 9 \div 6 = 7.5

Final answer

(i)Triangle ADE is similar to triangle ABC (AA).

(ii)BC = 12 cm

(iii)AC = 7.5 cm

Your Turn 11.8

DE is parallel to BC. Find $x$ (the length BC) and the length AC.

(i)Explain why triangle ADE is similar to triangle ABC.
(ii)BC
(iii)AC

Diagram for Your Turn 11.8

Show worked solutionHide worked solution

(i)

Angle A is common to both triangles.

shared angle

Angle ADE = angle ABC.

corresponding angles, DE // BC

Angle AED = angle ACB.

corresponding angles, DE // BC

So triangle ADE is similar to triangle ABC.

AA: equal angles

(ii)

\displaystyle AB = 4 + 6 = 10

AD + DB

\displaystyle \dfrac{DE}{BC} = \dfrac{AD}{AB} \ \Rightarrow \ \dfrac{6}{x} = \dfrac{4}{10}

corresponding sides of similar triangles are in the same ratio

\displaystyle x = 6 \times 10 \div 4 = 15

(iii)

\displaystyle \dfrac{AE}{AC} = \dfrac{AD}{AB} \ \Rightarrow \ \dfrac{3}{AC} = \dfrac{4}{10}

corresponding sides of similar triangles are in the same ratio

\displaystyle AC = 3 \times 10 \div 4 = 7.5

Final answer

(i)Triangle ADE is similar to triangle ABC (AA).

(ii)BC = 15 cm

(iii)AC = 7.5 cm

Example 11.9

Are two rectangles similar?

Are the two rectangles in each part similar? Give a reason.

(i)A rectangle 12 cm by 8 cm and a rectangle 18 cm by 12 cm.
(ii)A rectangle 15 cm by 10 cm and a rectangle 12 cm by 6 cm.

Show solutionHide solution

Colour keyround, estimate or compare

(i)

All the angles of a rectangle are right angles, so the angles always match.

angles are equal automatically

\displaystyle \dfrac{18}{12} = \frac{3}{2}, \quad \dfrac{12}{8} = \frac{3}{2}

compare the ratios of the lengths and of the widths

(ii)

All the angles of a rectangle are right angles, so the angles always match.

angles are equal automatically

\displaystyle \dfrac{12}{15} = \frac{4}{5}, \quad \dfrac{6}{10} = \frac{3}{5}

compare the ratios of the lengths and of the widths

Final answer

(i)similar (the ratios are equal)

(ii)not similar (the ratios are different)

Your Turn 11.9

Are the two rectangles in each part similar? Give a reason.

(i)A rectangle 20 cm by 15 cm and a rectangle 12 cm by 9 cm.
(ii)A rectangle 10 cm by 4 cm and a rectangle 15 cm by 8 cm.

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(i)

All the angles of a rectangle are right angles, so the angles always match.

angles are equal automatically

\displaystyle \dfrac{12}{20} = \frac{3}{5}, \quad \dfrac{9}{15} = \frac{3}{5}

compare the ratios of the lengths and of the widths

(ii)

All the angles of a rectangle are right angles, so the angles always match.

angles are equal automatically

\displaystyle \dfrac{15}{10} = \frac{3}{2}, \quad \dfrac{8}{4} = 2

compare the ratios of the lengths and of the widths

Final answer

(i)similar (the ratios are equal)

(ii)not similar (the ratios are different)

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Notes version 1.0 · Last updated 3 Nov 2026 · Chapter 11 of 19