Similar shapes

In a nutshell

Similar shapes are shapes that are the same but different in size. This difference in size is based on a common scale factor. Shapes have to abide by at least one condition to prove they are similar. 

Similarity

Two shapes are similar if all the angles are the same or all the sides have been enlarged or reduced by the same scale factor. 

Example 1

Show that the two triangles below are similar.

Both triangles have a right-angle, and a common angle of $$30^\circ$$. 

Find the missing angles using the fact that angles in a triangle add up to $$180^\circ$$.  

 $$180-(30+90)=180-(120)=60^\circ$$.

The angles in both triangles are $$90^\circ, \ 30^\circ$$ and $$60^\circ$$

The triangles are similar by AAA.

Note: The $$AAA$$ rule means all three angles in a triangle are the same. 

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Using scale factors

When two shapes are similar, it means the corresponding side lengths are proportional to one another. This means that each side in one shape can be multiplied by one number to give the corresponding side lengths in a similar shape. This number is the scale factor. 

Example 2

The two triangles shown below are similar. What is the length of the missing side?

Similar triangles will have a scale factor between them.

Find the scale factor by comparing the corresponding sides. 

$$6 \div 3 = 2$$​​​

The scale factor to go from the smaller to larger triangle is $$2$$. 

The missing side is corresponding with the side with length $$5$$. 

$$5 \times 2 = 10$$​​​

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Therefore, the missing side is $$\underline{10}$$ units long.