G19, R12 - Congruence and similarity of lengths

Congruent figures have the same size and shape; similar figures keep the same shape while allowing a change of size. You will match corresponding sides, use length scale factors in either direction, and explain why similar right triangles have equal side ratios.

Congruent and similar figures

A photo is $6\,\mathrm{cm}$ wide and $4\,\mathrm{cm}$ high. If you print it $9\,\mathrm{cm}$ wide, what height keeps the picture from looking stretched? Matching just the width is not enough: every corresponding length needs the same multiplier.

You already know

Translation, rotation and reflection preserve lengths and angles. An enlargement multiplies every length by its scale factor. A triangle's angles sum to $180^\circ$ .

Two figures can face different ways and still match. Turn or reflect them mentally before deciding which vertices correspond.

Congruence and similarity

Congruent figures fit exactly after a combination of translations, rotations or reflections: corresponding lengths and angles are equal.

Similar figures have equal corresponding angles and corresponding lengths in one common ratio. Congruent figures are the scale-factor- $1$ case of similarity.

Corresponding means matching positions. If $ABC$ is similar to $PQR$ in that order, $A$ matches $P$ , $B$ matches $Q$ , and $C$ matches $R$ . Thus $AB$ matches $PQ$ , not whichever side happens to be nearest it on the page.

Recall the triangle congruence tests: SSS, SAS with the included angle, ASA, and RHS with a right angle, hypotenuse and another side. For triangle similarity, use:

Test	What must match
AA	Two corresponding angles; the third then follows from the angle sum
SSS similarity	All three corresponding side ratios
SAS similarity	Two corresponding side ratios and their equal included angle

These criteria are not on the formula sheet: learn them. For example, two angles of $50^\circ$ and $70^\circ$ fix the third at $60^\circ$ , whatever the triangle's size.

Worked example 1: a common side multiplier

Question. Are triangles with sides $3,4,5\,\mathrm{cm}$ and $6,8,10\,\mathrm{cm}$ similar? Are they congruent?

Plan. Match sides by length order and compare all three ratios.

\begin{aligned} \frac63&=2 &&\text{shortest pair}\\ \frac84&=2 &&\text{middle pair}\\ \frac{10}5&=2 &&\text{longest pair} \end{aligned}

Why this step? One matching ratio does not establish that the whole shape has been scaled consistently.

Answer. They are similar by SSS similarity. They are not congruent because their corresponding lengths differ.

Check. Every larger side is twice its partner, while the factor is not $1$ .

Change just the longest side of the second triangle, from $10$ to $11\,\mathrm{cm}$ . This tests whether two correct ratios are enough.

Worked example 2: one mismatched side

Question. Are triangles with sides $3,4,5\,\mathrm{cm}$ and $6,8,11\,\mathrm{cm}$ similar?

Plan. Compare the third side ratio with the first two.

\begin{aligned} \frac63&=\frac84=2 &&\text{two ratios agree}\\ \frac{11}5&=2.2 &&\text{third ratio differs} \end{aligned}

Why this step? Similarity needs the same multiplier for every corresponding length; two equal ratios do not make the third one equal.

Answer. The triangles are not similar.

Check. Doubling the first triangle's longest side gives $10\,\mathrm{cm}$ , not $11\,\mathrm{cm}$ .

AA is special to triangles. Rectangles of dimensions $6$ by $3\,\mathrm{cm}$ and $8$ by $3\,\mathrm{cm}$ both have four right angles, but their length ratios are $8/6=4/3$ and $3/3=1$ . Equal angles alone do not make these rectangles similar.

Watch out

Wrong: “Both rectangles have right angles, so they are similar.” Their widths changed by a different factor from their heights. Check both dimensions; these rectangles are not similar.

Now change just one side in the second triangle again.

Scale factors and corresponding lengths

A scale factor has a direction: divide a destination length by its corresponding starting length.

For the opening photo, width changes by $9/6=1.5$ . The height must change by that same factor, from $4$ to $6\,\mathrm{cm}$ . Adding $3\,\mathrm{cm}$ to both dimensions would distort it.

Corresponding-length scale factor

k=\frac{\text{destination length}}{\text{starting length}}\qquad \text{destination length}=k\times\text{starting length}

Use matching lengths in the same unit. Reversing direction uses $1/k$ . Perimeters also multiply by $k$ , because every boundary segment does.

Not on the formula sheet: learn it.

The similar triangles below show one factor applying to all three corresponding lengths.

[DIAGRAM: asset_name: Corresponding lengths in similar right triangles; asset_slug: 1ma1_m57_similar_triangles; caption: The same multiplier takes every smaller length to its corresponding larger length; description: A blue triangle ABC has sides AB 6 cm, BC 8 cm and AC 10 cm. A sage similar triangle PQR has sides PQ 9 cm, QR 12 cm and PR 15 cm. Matching vertices and right angles are labelled, with a caption showing scale factor 1.5 from ABC to PQR.]
The same multiplier takes every smaller length to its corresponding larger length

Worked example 3: scaling towards the larger triangle

Question. Similar triangles $ABC$ and $PQR$ correspond in that order. $AB=6$ , $BC=8$ , $AC=10$ , $PQ=9$ , all in centimetres. Find $QR$ , $PR$ and the larger perimeter.

Plan. Find the factor from $ABC$ to $PQR$ , then multiply each matching length.

\begin{aligned} k&=\frac96=1.5 &&\text{destination over start}\\ QR&=8(1.5)=12\qquad PR=10(1.5)=15 &&\text{matching sides}\\ P_{PQR}&=(6+8+10)(1.5)=36\,\mathrm{cm} &&\text{perimeter is a length} \end{aligned}

Why this step? The stated vertex order fixes which sides correspond before any ratio is formed.

Answer. $QR=12\,\mathrm{cm}$ , $PR=15\,\mathrm{cm}$ , perimeter $36\,\mathrm{cm}$ .

Check. $9+12+15=36$ , and each side ratio is $1.5$ .

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G19, R12 - Congruence and similarity of lengths

Congruent and similar figures

Scale factors and corresponding lengths

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