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.

picture of 3 similar triangles

What is true about the angles of similar triangles?

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

angle of similar triangle

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 ..

angle of similar triangle

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

angle of similar triangle

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

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
picture of similar triangles 2

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

unsolvable triangle

Practice Problems

Problem 1

If $$ \triangle $$ ABC ~ $$\triangle $$ADE , AB = 20 and AD = 30, what is the similarity ratio?

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)

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

Two Similar Triangles

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 $

Problem 2

Use your knowledge of similar triangles to find the side lengths below.

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.

Problem 3

Below are two different versions of $$\triangle $$ HYZ and $$\triangle $$ HIJ . The only difference between the version is how long the sides are.

Only one of these two versions includes a pair of similar triangles.

Can you identify which version represents similar triangles?

identify similar ratios