How do you use similarity to find the length of a missing side?

Compare corresponding lengths to find a scale factor, then use the scale factor to find the missing length.

What is the condition to prove that two triangles are similar?

To prove two triangles are similar, use AAA - three matching angles.

What does it mean for two shapes to be similar?

Two shapes are similar if they have the same shape but have scaled side lengths.

Unlimited power

Get the whole package!

Learn without limits

Explainer videos without evulpo advertising

Unlimited number of exercises

Printable summaries

Learn without limits

Weekly learning report

Cancel any time

From £ 6.67 /month

Benefit from a yearly subscription now!

Save 44%

~~£11.99 / Month~~

Save 44%

£6.67 / Month

Chapter overview

Learning goals

Maths

Similar shapes: Scaling

Your lesson progress

Summary

Download

Similar shapes: Scaling

In a nutshell

Similar shapes are shapes that are the same shape, but have different sizes. There is a single condition to prove two shapes are similar. Similar shapes have a scale factor which can be used to find missing lengths.

Similarity

Two shapes are similar if they have the same angles. This means that the condition to prove that two triangles are similar is AAA - three common angles. Two similar shapes can have different side lengths that are in proportion to one another.

Example 1

Show that the two triangles below are similar.

Maths; Shapes and area; KS4 Year 10; Similar shapes: Scaling

Both triangles have a right-angle, and a common angle of $30^\circ$ . This means they also have a common angle of $180-(30+90)=60^\circ$ .

The triangles are similar due to AAA.

Similarity and scaling

When two shapes are similar, it means the matching side lengths are proportional to one another. This means that the side lengths of one shape can be multiplied by a scale factor to give the side lengths of another similar shape.

Example 2

The two triangles shown below are similar. What is the value of $x$ ?

If the two triangles are similar, then there must be a common scale factor.

Find the scale factor by comparing the lengths $RM$ and $LS$ :

$\frac{LS}{RM}=\frac{10}{5}=2$ 

The scale factor to go from triangle $AMR$ to triangle $LOS$ is therefore $2$ .

Use this scale factor to find the value of $x$ by focusing on the lengths $AR$ and $LO$ :

$LO=2\times AR$ 

$3x+3=2\times (x+2)$ 

$3x+3=2x+4$ 

$x=1$ 

Therefore, $\underline{ x=1}$ .

Similar shapes: Scaling

In a nutshell

Similarity

Example 1

Show that the two triangles below are similar.

Both triangles have a right-angle, and a common angle of $30^\circ$ . This means they also have a common angle of $180-(30+90)=60^\circ$ .

The triangles are similar due to AAA.

Similarity and scaling

Example 2

The two triangles shown below are similar. What is the value of $x$ ?

If the two triangles are similar, then there must be a common scale factor.

Find the scale factor by comparing the lengths $RM$ and $LS$ :

$\frac{LS}{RM}=\frac{10}{5}=2$ 

The scale factor to go from triangle $AMR$ to triangle $LOS$ is therefore $2$ .

Use this scale factor to find the value of $x$ by focusing on the lengths $AR$ and $LO$ :

$LO=2\times AR$ 

$3x+3=2\times (x+2)$ 

$3x+3=2x+4$ 

$x=1$ 

Therefore, $\underline{ x=1}$ .

Want to find out more? Check out these other lessons!

Similar shapes

Go to lesson

Frequently Asked Questions (FAQ)

FAQs

Question: How do you use similarity to find the length of a missing side?

Answer: Compare corresponding lengths to find a scale factor, then use the scale factor to find the missing length.
Question: What is the condition to prove that two triangles are similar?

Answer: To prove two triangles are similar, use AAA - three matching angles.
Question: What does it mean for two shapes to be similar?

Answer: Two shapes are similar if they have the same shape but have scaled side lengths.

Theory

Exercises

Your data protection

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information that you’ve provided them or that they’ve collected from your use of their services. By clicking on either "Accept cookies" or "Necessary cookies only", you agree to this (read more in our Privacy Policy). Privacy Policy

Unlimited power

Learn without limits

Benefit from a yearly subscription now!

Similar shapes: Scaling

Similar shapes: Scaling

In a nutshell

Similarity

Example 1

Similarity and scaling

Example 2

Similar shapes: Scaling

In a nutshell

Similarity

Example 1

Similarity and scaling

Example 2

Want to find out more? Check out these other lessons!

Now you’ve read the summary, have a go at the exercises!