Corresponding lengths in similar figures | Year 9

01 / LEARN

Understand the idea.

Correspondence comes before calculation

Corresponding parts occupy matching roles in similar figures, even after a rotation or reflection. Corresponding angles are equal, and all corresponding lengths have the same scale factor. Position on the page is not a reliable way to match them.

Read a similarity statement in order. In $△ A B C \sim △ D E F$ , vertices $A, B, C$ match $D, E, F$ respectively. Match both endpoints to pair sides: $A B$ with $D E$ , $B C$ with $E F$ , and $A C$ with $D F$ .

Use one consistent direction

Choose a direction and label it before dividing. In the supplied figures, large divided by small gives $\frac{12}{8}$ , $\frac{9}{6}$ and $\frac{15}{10}$ . Mixing small-to-large with large-to-small ratios would give reciprocal scale factors.

A picture that looks similar is not enough. Use stated similarity or a valid triangle similarity test, such as three proportional corresponding sides. One equal pair of side ratios alone cannot establish that two arbitrary triangles have the same shape.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read a similarity statement

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

Worked solution

1
Match the vertices in their written order.
$K \leftrightarrow R, L \leftrightarrow S, M \leftrightarrow T$
The similarity statement specifies which angles and vertices correspond.
2
Match both endpoints of KL.
$K L R S$
K matches R and L matches S.
3
Match both endpoints of KM.
$K M R T$
K matches R and M matches T, regardless of the triangles' orientation.

Answer

KL corresponds to RS; KM corresponds to RT.

EXAMPLE 02

Check proportional sides

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

Worked solution

1
Compare the corresponding sides DE and AB.
$\frac{D E}{A B} = \frac{12}{8} = 1.5$
Large-to-small is the chosen comparison direction.
2
Compare DF and AC in the same direction.
$\frac{D F}{A C} = \frac{9}{6} = 1.5$
The endpoints determine this second corresponding pair.
3
Compare EF and BC.
$\frac{E F}{B C} = \frac{15}{10} = 1.5$
The third side pair has the same multiplier, so all three requested ratios equal one and a half.

Answer

All three ratios equal 1.5.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Angle Y

The order in $△ P Q R \sim △ X Y Z$ pairs $P$ with $X$ , $Q$ with $Y$ , and $R$ with $Z$ . Therefore angle $Q$ corresponds to angle $Y$ .

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6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

If $△ A B C \sim △ K L M$ , name the side corresponding to $A C$ .
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 $\frac{A B}{D E} = \frac{B C}{D F}$ 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 $A C$ .
1 mark
$A$ matches $K$ and $C$ matches $M$ , so $A C$ matches $K M$ .
$K M$
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
$\frac{6}{4} = \frac{9}{6} = \frac{12}{8} = 1.5$ .
$1.5$
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 $\frac{A B}{D E} = \frac{B C}{D F}$ 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 $\frac{A B}{D E} = \frac{B C}{E F}$ .
$B C$ must be paired with $E F$ , not $D F$ .
Can sides $(3, 4, 5)$ and $(6, 8, 11)$ be corresponding sides of similar triangles?
2 marks
$\frac{6}{3} = 2$ and $\frac{8}{4} = 2$ , but $\frac{11}{5} = 2.2$ .
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 ↗

In this lesson

Understand the idea.

Correspondence comes before calculation

Use one consistent direction

See the method in action.

Read a similarity statement

Match the vertices in their written order.

Match both endpoints of KL.

Match both endpoints of KM.

Check proportional sides

Compare the corresponding sides DE and AB.

Compare DF and AC in the same direction.

Compare EF and BC.

Try it yourself.

Check your understanding

Keep practising