# Similar Triangles

Angles, Sides & similarity ratio

#### What are similar triangles?

Answer: Similar triangles have the same 'shape' but are just scaled differently. Similar triangles have congruent angles and proportional sides.

#### What is true about the angles of similar triangles?

Answer: They are congruent. as the picture below demonstrates.

#### What is true about the sides of similar triangles?

Answer: Corresponding sides of similar triangles are proportional. The example below shows two triangle's with their proportional sides ..

#### What is the similarity ratio (aka scale factor)?

Answer: It's the ratio between corresponding sides. In the picture above, the larger triangle's sides are two times the smaller triangles sides so the scale factor is 2

$$16 \cdot 2 = 32 \\ 22 \cdot 2 = 44 \\ 25 \cdot 2 = 50$$

Notation: $$\triangle ABC$$~$$\triangle XYZ$$ means that "$$\triangle ABC \text{ is similar to } \triangle XYZ$$"

#### How do you find the similarity ratio?

Answer: Match up any pair of corresponding sides and set up a ratio. That's it!

If $$\triangle ABC$$ ~ $$\triangle WXY$$, then what is the similarity ratio?

Step 1

Pick a pair of corresponding sides
(follow the letters)

AB and WX are corresponding.

Follow the letters: $$\triangle \color{red}{AB}C$$ ~ $$\triangle \color{red}{WX}Y$$

Step 2

Substitute side lengths into proportion

$$\frac{AB}{WX} = \frac{7}{21}$$

Step 3

Simplify (if necessary)

$$\frac{7}{21}=\frac{1}{3}$$

Step 3

#### Why is the following problem unsolvable?

If $$\triangle$$ JKL ~ $$\triangle$$ XYZ, LJ = 22 ,JK = 20 and YZ = 30, what is the similarity ratio?

Answer: You are not given a single pair of corresponding sides so you cannot find the similarity ratio.

Remember: How to Find corresponding sides

Corresponding sides follow the same letter order as the triangle name so:

• YZ of $$\triangle X\color{red}{YZ}$$ corresponds with side KL of$$\triangle J\color{red}{KL}$$
• JK of $$\triangle \color{red}{JK}L$$ corresponds with side XY of$$\triangle \color{red}{XY}Z$$
• LJ of $$\triangle \color{red}{J}K\color{red}{L}$$ corresponds with side ZX of$$\triangle \color{red}{X}Y \color{red}{Z}$$

Below is a picture of what these two triangles could look like

### Practice Problems

Step 1

Pick a pair of corresponding sides (follow the letters)

AB and AD are corresponding based on the letters of the triangle names
$$\triangle \color{red}{AB}C$$ ~ $$\triangle \color{red}{AD}E$$

Step 2

Substitute side lengths into proportion

$$\frac{AB}{AD} = \frac{20}{30}$$

Step 3

Simplify (if necessary)

$$\frac{20}{30} = \frac{2}{3}$$

Part B) If EA = 33, how long is CA?

EA and CA are corresponding sides ( $$\triangle \color{red}{A}B\color{red}{C}$$ ~ $$\triangle \color{red}{A}D\color{red}{E}$$ )

Since the sides of similar triangles are proportional, just set up a proportion involving these two sides and the similarity ratio and solve.

$\frac{EA}{CA} = \frac{3}{2} \\ \frac{33}{CA} = \frac{3}{2} \\ CA \cdot 3 = 2 \cdot 33 \\ CA \cdot 3 = 66 \\ CA = \frac{66}{3} = 22$

DE = 27, how long is BC?

EA and AC are corresponding sides ($$\triangle \color{red}{ A}B\color{red}{C}$$ ~ $$\triangle \color{red}{A}D\color{red}{E}$$)

Since the sides of similar triangles are proportional, just set up a proportion involving these two sides and the similarity ratio and solve.

$\frac{DE}{BC} = \frac{3}{2} \\ \frac{27}{CA} = \frac{3}{2} \\ CA \cdot 3 = 2 \cdot 27 \\ CA \cdot 3 = 54 \\ CA = \frac{54}{3} = 18$

Step 1

Pick a pair of corresponding sides (follow the letters)

HY and HI are corresponding sides

$$\triangle \color{red}{HY}Z$$ ~ $$\triangle \color{red}{HI}Y$$

Step 2

Substitute side lengths into proportion

$$\frac{HY}{HI } = \frac{8}{12}$$

(You could, of course, have flipped this fraction if you wanted to put HI in the numerator $$\frac{HI}{HY}$$ )
Step 3

Simplify (if necessary)

$$\frac{8}{12}=\frac{2}{3}$$

Step 4

Set up equation involving ratio and a pair of corresponding sides

$$\frac{2}{3} =\frac{YZ}{IJ} \\ \frac{2}{3} =\frac{YZ}{9} \\ \frac{2 \cdot 9}{3} =YZ \\ YZ = 6$$

Finding ZJ is a bit more tricky . You could use the side splitter short cut . Or you use the steps up above to find the length of HJ ,which is 6 and then subtract HZ (or 4) from that to get the answer.