Corresponding lengths in similar figures | Year 9

01 / LEARN

The idea, step by step

Correspondence comes before calculation

Similar figures have equal corresponding angles and proportional corresponding side lengths. The word corresponding means playing the same geometric role, not simply being in the same place on the page. A rotated or reflected copy may look different.

The statement $△ A B C \sim △ D E F$ gives the matching order: $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ . Therefore AB matches DE, BC matches EF and AC matches DF.

Use one consistent direction

Choose a direction such as small to large, and keep it in every fraction. In the diagram $D E / A B = 12 / 8 = 1.5$ , $D F / A C = 9 / 6 = 1.5$ and $E F / B C = 15 / 10 = 1.5$ .

One equal side ratio alone is not sufficient to establish that arbitrary triangles are similar. Use stated similarity, matching angles or a valid triangle similarity test. A drawing that looks similar is not evidence.

A common trap

The longest side always corresponds to the bottom side in the next diagram.

Instead: Orientation can change. Use vertex/angle correspondence; corresponding longest sides match each other, wherever they are drawn.

02 / WORKED EXAMPLES

See the thinking

Example 1

Read a similarity statement

If $△ K L M \sim △ R S T$ , which sides correspond to KL and KM?

Match letters in order: $K \leftrightarrow R$ , $L \leftrightarrow S$ , $M \leftrightarrow T$ .
The endpoints K,L match R,S; endpoints K,M match R,T.

KL corresponds to RS; KM corresponds to RT.

Example 2

Check proportional sides

Use the supplied triangles to check all three large-to-small length ratios.

$D E / A B = 12 / 8 = 1.5$ .
$D F / A C = 9 / 6 = 1.5$ .
$E F / B C = 15 / 10 = 1.5$ .

All three ratios equal 1.5.

Check your understanding

If $△ P Q R \sim △ X Y Z$ , which angle matches angle Q?

Show answer

Angle Y

The second letter Q corresponds to the second letter Y.

03 / WORKSHEET

Your turn

6 questions · 11 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

If $△ A B C \sim △ K L M$ , name the side corresponding to AC.
1 mark
If $△ A B C \sim △ K L M$ , name the angle corresponding to B.
1 mark
Triangles have corresponding side lengths $(4, 6, 8)$ and $(6, 9, 12)$ . Find the scale factor from first to second.
2 marks
Two triangles have angles $(35^{\circ}, 65^{\circ}, 80^{\circ})$ and $(80^{\circ}, 35^{\circ}, 65^{\circ})$ . Are they similar? Explain.
2 marks
A learner uses $A B / D E = B C / D F$ for $△ A B C \sim △ D E F$ . Explain the mismatch.
3 marks
Can sides $(3, 4, 5)$ and $(6, 8, 11)$ be corresponding sides of similar triangles?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

If $△ A B C \sim △ K L M$ , name the side corresponding to AC.
1 mark
A matches K and C matches M, so AC matches KM.
KM
If $△ A B C \sim △ K L M$ , name the angle corresponding to B.
1 mark
B is the second letter; L is the second letter.
Angle L
Triangles have corresponding side lengths $(4, 6, 8)$ and $(6, 9, 12)$ . Find the scale factor from first to second.
2 marks
$6 / 4 = 9 / 6 = 12 / 8 = 1.5$ .
1.5
Two triangles have angles $(35^{\circ}, 65^{\circ}, 80^{\circ})$ and $(80^{\circ}, 35^{\circ}, 65^{\circ})$ . Are they similar? Explain.
2 marks
Their three corresponding angles are equal, despite a different listed order.
The AA angle test establishes similarity.
Yes.
A learner uses $A B / D E = B C / D F$ for $△ A B C \sim △ D E F$ . Explain the mismatch.
3 marks
The statement matches B to E and C to F.
A consistent ratio is $A B / D E = B C / E F$ .
BC must be paired with EF, not DF.
Can sides $(3, 4, 5)$ and $(6, 8, 11)$ be corresponding sides of similar triangles?
2 marks
$6 / 3 = 2$ and $8 / 4 = 2$ , but $11 / 5 = 2.2$ .
There is no single scale factor.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 proportional modelling, rates, scale, similarity and enlargement. Assumptions, interpretation and model limitations are taught alongside calculation.

Official curriculum source ↗

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