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.

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.

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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, $$\underline{ x=1}$$.