Similar Shapes (Lengths)

What you'll learn

Recognise when two shapes are mathematically similar.
Match corresponding sides and use a scale factor.
Use parallel lines to spot similar triangles.
Prove when two rectangles are not similar.

Same shape, different size

Before calculating, you need to know what “similar” really means. Similar shapes may be different sizes, but they have exactly the same shape.

Definition

Mathematically similar shapes

Two shapes are mathematically similar if one is an enlargement of the other: all matching angles are equal, and all matching lengths are multiplied by the same scale factor. An enlargement means every length has been multiplied by one fixed number; the scale factor is that multiplier.

Definition

Corresponding sides

A vertex is a corner. Corresponding sides are matching sides in the same positions, usually found by matching the vertices first.

Key Idea

Length rule

In similar shapes, every pair of corresponding lengths has the same multiplier. If you know one matching pair, you can use it to find all the other missing lengths.

Example

Finding missing lengths in similar triangles

Two triangles, $A B C$ and $P QR$ , are similar. $A$ matches $P$ , $B$ matches $Q$ , so $C$ matches $R$ . $A B = 10$ cm, $A C = 24$ cm, $P Q = 15$ cm and $QR = 27$ cm. Find $P R$ and $B C$ .

Two similar triangles with corresponding vertices in the same order, showing the known matching sides and the two missing lengths.

Match the corresponding sides: $\leftrightarrow PQ$ , $\leftrightarrow PR$ , and $\leftrightarrow QR$ .
Let $k$ be the scale factor from triangle $A B C$ to triangle $P QR$ : $k=1510=1.5k=\frac{15}{10}=1.5$ .
Since $\leftrightarrow PR$ , use $\times 1.5 = 36$ . So $P R$ is 36 cm.
Since $\leftrightarrow QR`, use$ BC = 27 \div 1.5 = 18 $. S o$ BC$ is 18 cm.

Common Mistake

Using the wrong matching side

Do not compare two sides just because their numbers are given. Compare sides in the same positions, or sides between matching angle labels.

Parallel lines inside triangles

Parallel lines run in the same direction and never meet. In GCSE similarity questions, parallel lines often create similar triangles.

If a line inside a triangle is parallel to one side, it creates a smaller triangle similar to the whole triangle.

A line drawn parallel to one side of a triangle creates a smaller triangle inside the larger one.

Example

Triangle with a parallel line

In triangle $A C D$ , point $B$ lies on $A C$ , point $E$ lies on $A D$ , and $B E$ is parallel to $C D$ . $A B = 6$ cm, $B C = 4$ cm, $A E = 7.5$ cm and $C D = 15$ cm. Find $E D$ and $B E$ .

Triangle ACD with BE parallel to CD, showing the split sides and the missing parallel length.

Add along the straight side to get the whole length: $A C = 6 + 4 = 10$ .
The small triangle $A B E$ matches the whole triangle $A C D$ . Let $k$ be the scale factor from small to whole: $k=ACAB=106=53k=\frac{AC}{AB}=\frac{10}{6}=\frac{5}{3}$ .
Since $\leftrightarrow AD$ , find the whole side: $\times \frac{5}{3}=12.5$ .
The question asks for the extra part: $E D = A D - A E = 12.5 - 7.5 = 5$ , so $E D$ is 5 cm.
Since $\leftrightarrow CD$ , use $\div \frac{5}{3}=9$ , so $B E$ is 9 cm.

Common Mistake

Part or whole?

If a side has been split, be clear whether you need the small part, the extra part, or the whole side. For example, $A C = A B + B C$ , but $E D = A D - A E$ .

Crossed lines between parallel sides

Sometimes the triangles are on opposite sides of a crossing point. The same idea still works: the two parallel sides correspond, and the sloping sides match up through the crossing point.

Example

Crossing lines with parallel sides

$M N$ is parallel to $S T$ . The lines $M T$ and $N S$ cross at $P$ . $M N = 8$ cm, $S T = 20$ cm and $P T = 15$ cm. Find $M P$ .

Crossing lines form two similar triangles on opposite sides of P, with MN parallel to ST.

The two similar triangles are $M N P$ and $T S P$ .
Match the parallel sides: $\leftrightarrow ST$ , so the scale factor from $M N P$ to $T S P$ is $k=208=2.5k=\frac{20}{8}=2.5$ .
The matching sloping sides are $\leftrightarrow PT$ .
$P T$ is on the larger triangle, so divide by the scale factor: $\div 2.5 = 6$ . So $M P$ is 6 cm.

Tip

Sanity check

If the parallel side on one triangle is larger, every corresponding length on that triangle should be larger by the same scale factor.

Similar triangles can face opposite ways

In a “bow-tie” style diagram, two triangles meet at one point and face away from each other. They can still be similar if they share angles and have a pair of parallel sides.

Example

Two triangles meeting at one point

$A B$ is parallel to $D E$ . Points $A$ , $C$ , $E$ lie on one straight line, and $B$ , $C$ , $D$ lie on another. $A B = 4$ cm, $D E = 6$ cm, $A C = 5$ cm and $C D = 12$ cm. Find $C E$ and $B C$ .

A bow-tie pair of similar triangles meeting at C, with AB parallel to DE.

The similar triangles are $A B C$ and $E D C$ .
Match the parallel sides: $\leftrightarrow DE$ , so $k=64=1.5k=\frac{6}{4}=1.5$ from $A B C$ to $E D C$ .
$\leftrightarrow CE$ , so $\times 1.5 = 7.5$ . Therefore $C E$ is 7.5 cm.
$\leftrightarrow CD$ , so $\div 1.5 = 8$ . Therefore $B C$ is 8 cm.

Using ratios in similar shapes

A ratio compares two amounts. A ratio of 1:3 means the second amount has 3 equal parts for every 1 equal part of the first.

A right-angled triangle has one 90° angle. If two right-angled triangles share another angle as well, they are similar.

Two right-angled triangles sharing angle A are similar because they have two equal angles.

Example

Using a length ratio

A small right-angled triangle $A B E$ sits inside a larger right-angled triangle $A C D$ . They share angle $A$ . $A B = 6$ cm, $B E = 5$ cm and $A B : A C = 2 : 5$ . Find $C D$ and $B C$ .

Nested right-angled triangles sharing angle A, with the base ratio giving the scale factor from small to large.

The ratio $A B : A C = 2 : 5$ means the scale factor from the small triangle to the large triangle is $k=52=2.5k=\frac{5}{2}=2.5$ .
$B E$ corresponds to $C D$ , so $\times 2.5 = 12.5`. Therefore$ CD$ is 12.5 cm.
$A B$ corresponds to $A C$ , so $\times 2.5 = 15$ .
$B C$ is the remaining part of $A C$ : $B C = 15 - 6 = 9$ . Therefore $B C$ is 9 cm.

Tip

Ratios are scale factors in disguise

For $A B : A C = 2 : 5$ , think “small to large is 2 parts to 5 parts”, so multiply by $52\frac{5}{2}$ to go from $A B$ to $A C$ .

Showing shapes are not similar

For rectangles, all angles are right angles, so the angle condition is already satisfied. To prove two rectangles are similar, the length scale factor and the width scale factor must be the same.

Example

Showing two rectangles are not similar

Rectangle A is 120 mm long and 70 mm wide. Rectangle B is 150 mm long and 90 mm wide. Show they are not mathematically similar.

The two rectangles have different length and width scale factors, so they are not similar.

Compare the lengths. Let $klengthk_{\text{length}}$ be the scale factor from A to B: $klength=150120=1.25k_{\text{length}}=\frac{150}{120}=1.25$ .
Compare the widths. Let $kwidthk_{\text{width}}$ be the scale factor from A to B: $kwidth=9070≈1.29k_{\text{width}}=\frac{90}{70}\approx1.29$ .
The scale factors are different, so the same multiplier has not been used for both measurements. The rectangles are not mathematically similar.

Common Mistake

Being a rectangle is not enough

All rectangles have matching angles, but their length-to-width ratios can be different. You must check both length and width scale factors.

Exam technique

In the exam

Mark corresponding sides on the diagram before calculating.
Use one complete matching pair to find the scale factor.
Check whether you need a whole side or just a missing section.

Self review

Check yourself

Can you identify corresponding sides when two similar triangles face opposite ways?
If $A B : A C = 1 : 3$ , what scale factor takes $A B$ to $A C$ ?
For two rectangles, how can you prove they are not mathematically similar?

Similar Shapes (Lengths)

What you'll learn

Recognise when two shapes are mathematically similar.
Match corresponding sides and use a scale factor.
Use parallel lines to spot similar triangles.
Prove when two rectangles are not similar.

Same shape, different size

Before calculating, you need to know what “similar” really means. Similar shapes may be different sizes, but they have exactly the same shape.

Definition

Mathematically similar shapes

Definition

Corresponding sides

A vertex is a corner. Corresponding sides are matching sides in the same positions, usually found by matching the vertices first.

Key Idea

Length rule

In similar shapes, every pair of corresponding lengths has the same multiplier. If you know one matching pair, you can use it to find all the other missing lengths.

Example

Finding missing lengths in similar triangles

Two triangles, $A B C$ and $P QR$ , are similar. $A$ matches $P$ , $B$ matches $Q$ , so $C$ matches $R$ . $A B = 10$ cm, $A C = 24$ cm, $P Q = 15$ cm and $QR = 27$ cm. Find $P R$ and $B C$ .

Two similar triangles with corresponding vertices in the same order, showing the known matching sides and the two missing lengths.

Match the corresponding sides: $\leftrightarrow PQ$ , $\leftrightarrow PR$ , and $\leftrightarrow QR$ .
Let $k$ be the scale factor from triangle $A B C$ to triangle $P QR$ : $k=1510=1.5k=\frac{15}{10}=1.5$ .
Since $\leftrightarrow PR$ , use $\times 1.5 = 36$ . So $P R$ is 36 cm.
Since $\leftrightarrow QR`, use$ BC = 27 \div 1.5 = 18 $. S o$ BC$ is 18 cm.

Common Mistake

Using the wrong matching side

Do not compare two sides just because their numbers are given. Compare sides in the same positions, or sides between matching angle labels.

Parallel lines inside triangles

Parallel lines run in the same direction and never meet. In GCSE similarity questions, parallel lines often create similar triangles.

If a line inside a triangle is parallel to one side, it creates a smaller triangle similar to the whole triangle.

A line drawn parallel to one side of a triangle creates a smaller triangle inside the larger one.

Example

Triangle with a parallel line

In triangle $A C D$ , point $B$ lies on $A C$ , point $E$ lies on $A D$ , and $B E$ is parallel to $C D$ . $A B = 6$ cm, $B C = 4$ cm, $A E = 7.5$ cm and $C D = 15$ cm. Find $E D$ and $B E$ .

Triangle ACD with BE parallel to CD, showing the split sides and the missing parallel length.

Add along the straight side to get the whole length: $A C = 6 + 4 = 10$ .
The small triangle $A B E$ matches the whole triangle $A C D$ . Let $k$ be the scale factor from small to whole: $k=ACAB=106=53k=\frac{AC}{AB}=\frac{10}{6}=\frac{5}{3}$ .
Since $\leftrightarrow AD$ , find the whole side: $\times \frac{5}{3}=12.5$ .
The question asks for the extra part: $E D = A D - A E = 12.5 - 7.5 = 5$ , so $E D$ is 5 cm.
Since $\leftrightarrow CD$ , use $\div \frac{5}{3}=9$ , so $B E$ is 9 cm.

Common Mistake

Part or whole?

If a side has been split, be clear whether you need the small part, the extra part, or the whole side. For example, $A C = A B + B C$ , but $E D = A D - A E$ .

Crossed lines between parallel sides

Sometimes the triangles are on opposite sides of a crossing point. The same idea still works: the two parallel sides correspond, and the sloping sides match up through the crossing point.

Example

Crossing lines with parallel sides

$M N$ is parallel to $S T$ . The lines $M T$ and $N S$ cross at $P$ . $M N = 8$ cm, $S T = 20$ cm and $P T = 15$ cm. Find $M P$ .

Crossing lines form two similar triangles on opposite sides of P, with MN parallel to ST.

The two similar triangles are $M N P$ and $T S P$ .
Match the parallel sides: $\leftrightarrow ST$ , so the scale factor from $M N P$ to $T S P$ is $k=208=2.5k=\frac{20}{8}=2.5$ .
The matching sloping sides are $\leftrightarrow PT$ .
$P T$ is on the larger triangle, so divide by the scale factor: $\div 2.5 = 6$ . So $M P$ is 6 cm.

Tip

Sanity check

If the parallel side on one triangle is larger, every corresponding length on that triangle should be larger by the same scale factor.

Similar triangles can face opposite ways

In a “bow-tie” style diagram, two triangles meet at one point and face away from each other. They can still be similar if they share angles and have a pair of parallel sides.

Example

Two triangles meeting at one point

$A B$ is parallel to $D E$ . Points $A$ , $C$ , $E$ lie on one straight line, and $B$ , $C$ , $D$ lie on another. $A B = 4$ cm, $D E = 6$ cm, $A C = 5$ cm and $C D = 12$ cm. Find $C E$ and $B C$ .

A bow-tie pair of similar triangles meeting at C, with AB parallel to DE.

The similar triangles are $A B C$ and $E D C$ .
Match the parallel sides: $\leftrightarrow DE$ , so $k=64=1.5k=\frac{6}{4}=1.5$ from $A B C$ to $E D C$ .
$\leftrightarrow CE$ , so $\times 1.5 = 7.5$ . Therefore $C E$ is 7.5 cm.
$\leftrightarrow CD$ , so $\div 1.5 = 8$ . Therefore $B C$ is 8 cm.

Using ratios in similar shapes

A ratio compares two amounts. A ratio of 1:3 means the second amount has 3 equal parts for every 1 equal part of the first.

A right-angled triangle has one 90° angle. If two right-angled triangles share another angle as well, they are similar.

Two right-angled triangles sharing angle A are similar because they have two equal angles.

Example

Using a length ratio

A small right-angled triangle $A B E$ sits inside a larger right-angled triangle $A C D$ . They share angle $A$ . $A B = 6$ cm, $B E = 5$ cm and $A B : A C = 2 : 5$ . Find $C D$ and $B C$ .

Nested right-angled triangles sharing angle A, with the base ratio giving the scale factor from small to large.

The ratio $A B : A C = 2 : 5$ means the scale factor from the small triangle to the large triangle is $k=52=2.5k=\frac{5}{2}=2.5$ .
$B E$ corresponds to $C D$ , so $\times 2.5 = 12.5`. Therefore$ CD$ is 12.5 cm.
$A B$ corresponds to $A C$ , so $\times 2.5 = 15$ .
$B C$ is the remaining part of $A C$ : $B C = 15 - 6 = 9$ . Therefore $B C$ is 9 cm.

Tip

Ratios are scale factors in disguise

For $A B : A C = 2 : 5$ , think “small to large is 2 parts to 5 parts”, so multiply by $52\frac{5}{2}$ to go from $A B$ to $A C$ .

Showing shapes are not similar

For rectangles, all angles are right angles, so the angle condition is already satisfied. To prove two rectangles are similar, the length scale factor and the width scale factor must be the same.

Example

Showing two rectangles are not similar

Rectangle A is 120 mm long and 70 mm wide. Rectangle B is 150 mm long and 90 mm wide. Show they are not mathematically similar.

The two rectangles have different length and width scale factors, so they are not similar.

Compare the lengths. Let $klengthk_{\text{length}}$ be the scale factor from A to B: $klength=150120=1.25k_{\text{length}}=\frac{150}{120}=1.25$ .
Compare the widths. Let $kwidthk_{\text{width}}$ be the scale factor from A to B: $kwidth=9070≈1.29k_{\text{width}}=\frac{90}{70}\approx1.29$ .
The scale factors are different, so the same multiplier has not been used for both measurements. The rectangles are not mathematically similar.

Common Mistake

Being a rectangle is not enough

All rectangles have matching angles, but their length-to-width ratios can be different. You must check both length and width scale factors.

Exam technique

In the exam

Mark corresponding sides on the diagram before calculating.
Use one complete matching pair to find the scale factor.
Check whether you need a whole side or just a missing section.

Self review

Check yourself

Can you identify corresponding sides when two similar triangles face opposite ways?
If $A B : A C = 1 : 3$ , what scale factor takes $A B$ to $A C$ ?
For two rectangles, how can you prove they are not mathematically similar?

Similar Shapes (Lengths)

What you'll learn

Same shape, different size

Parallel lines inside triangles

Crossed lines between parallel sides

Similar triangles can face opposite ways

Using ratios in similar shapes

Showing shapes are not similar

Recap questions

You've reached the end

Similar Shapes (Lengths)

What you'll learn

Same shape, different size

Parallel lines inside triangles

Crossed lines between parallel sides

Similar triangles can face opposite ways

Using ratios in similar shapes

Showing shapes are not similar

Recap questions

You've reached the end

Similar Shapes (Lengths)

What you'll learn

Same shape, different size

Parallel lines inside triangles

Crossed lines between parallel sides

Similar triangles can face opposite ways

Using ratios in similar shapes

Showing shapes are not similar

Recap questions

You've reached the end

Similar Shapes (Lengths) Revision Guide

Similar Shapes (Lengths)

What you'll learn

Same shape, different size

Parallel lines inside triangles

Crossed lines between parallel sides

Similar triangles can face opposite ways

Using ratios in similar shapes

Showing shapes are not similar

Recap questions

You've reached the end

Similar Shapes (Lengths) Revision Guide