Similar figures

The same shape, a different size

A photograph and its enlargement have the same shape. A map and the land it shows have the same shape. So do a toy car and the real car. Each one is the other stretched or shrunk by the same amount in every direction. We say the two figures are similar.

Similar figures

Two figures with the same number of sides are similar when both of these are true:

1. Their corresponding angles are equal.
2. Their corresponding sides are in the same ratio.

That common ratio is the scale factor. If the scale factor is $1.5$, every side of the second figure is $1.5$ times its match in the first, and the angles do not change.

Some figures are always similar:

• Any two circles. A circle changes only in its radius.
• Any two squares. Every angle is $90^\circ$, and the sides of one are in the same ratio to the sides of the other.
• Any two equilateral triangles. Every angle is $60^\circ$, and all the sides of each are equal.

Why both conditions are needed

A square and a rectangle have the same angles, all $90^\circ$. But they are not similar, because their sides are not in one ratio. A square and a rhombus with the same side length have equal side ratios, but their angles differ, so they are not similar either.

Equal angles alone are not enough. Equal ratios alone are not enough. We need both.

Two rectangles

Is a rectangle $4$ cm by $6$ cm similar to a rectangle $6$ cm by $9$ cm?

1. Angles. All the angles are $90^\circ$, so they match.
2. Sides. $\dfrac{6}{4} = 1.5$ and $\dfrac{9}{6} = 1.5$. The ratios are equal.

So the two rectangles are similar, with scale factor $1.5$.

Now try a rectangle $6$ cm by $8$ cm. Here $\dfrac{6}{4} = 1.5$, but $\dfrac{8}{6}$ is about $1.33$. The ratios differ, so this rectangle is not similar to the first.

Similar triangles

For triangles we write the vertices in matching order. The statement $\triangle ABC \sim \triangle DEF$ says that $A$ matches $D$, $B$ matches $E$ and $C$ matches $F$. Then

In the picture, matching angles carry the same number of arcs. Matching sides lie opposite matching angles. The order of the letters matters: $\triangle ABC \sim \triangle FED$ would say something different.

For triangles, one of the two conditions is enough. If the angles are equal, the sides are sure to be in one ratio. You will prove this in the third course of this chapter.

Finding a side of a similar triangle

Triangle $ABC$ has $AB = 4$ cm, $BC = 6$ cm and $CA = 5$ cm. Triangle $DEF$ is similar to it, with $EF = 9$ cm. Find $DE$ and $FD$.

1. Match the sides. $\triangle ABC \sim \triangle DEF$, so $BC$ matches $EF$.
2. Find the scale factor. $\dfrac{EF}{BC} = \dfrac{9}{6} = 1.5$.
3. Multiply the other sides. $DE = 4 \times 1.5 = 6$ cm and $FD = 5 \times 1.5 = 7.5$ cm.

Check one more pair. $\dfrac{DE}{AB} = \dfrac{6}{4} = 1.5$. It agrees.

A method

1. Match the vertices, or match the sides: shortest to shortest, longest to longest.
2. Find the scale factor from one pair of matching sides.
3. Multiply every other side by the scale factor.
4. Check one more pair.

Practise with new numbers

Try it in Python

The program lists the sides of two triangles from shortest to longest, and divides each pair.

Common mistakes

• Matching the wrong sides. The shortest side of one triangle matches the shortest side of the other, not the first side you wrote down.
• Adding instead of multiplying. A scale factor multiplies every side. Adding the same length to each side changes the shape.
• Using one condition for a rectangle or another figure. Equal angles alone do not make two rectangles similar.
• Writing the letters in the wrong order. In $\triangle ABC \sim \triangle DEF$, the order of the letters says which vertex matches which.

Remember these

• Similar figures have equal matching angles and matching sides in one ratio.
• That ratio is the scale factor.
• Any two circles are similar, and so are any two squares and any two equilateral triangles.
• Match the shortest side to the shortest, then multiply by the scale factor.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A map has a scale of $1 : 50\,000$. Two towns are $4$ cm apart on the map. How far apart are they, in kilometres?

2. Are all rectangles similar to one another? Give an example.

3. $\triangle PQR \sim \triangle XYZ$, with $PQ = 5$ cm, $QR = 7$ cm and $XY = 10$ cm. Find $YZ$.

4. $\triangle ABC \sim \triangle DEF$, with $\angle A = 50^\circ$ and $\angle B = 60^\circ$. Find $\angle F$.

5. Why do we write the letters of similar triangles in matching order?