Application of Similar Triangles

Once you know that two triangles are similar, you can use the fact that their corresponding sides are proportional and their corresponding angles are congruent to solve problems.

Consider the triangles in the image below.

a. Prove that the triangles are similar.

The triangles are similar by $A A \sim$ because they have at least two pairs of congruent angles, i.e.,

$\angle A B C=\angle D E F=90^{\circ}, \angle B C A=\angle E F D=30^{\circ}, \angle C A B=\angle F D E=60^{\circ}$

b. Use the Pythagorean Theorem to find $D E$.

According to the Pythagorean Theorem, in $\triangle D E F$ :

$\begin{aligned}E F^{2}+D E^{2} &=F D^{2} \\(3 \sqrt{3})^{2}+D E^{2} &=6^{2} \\(9 \cdot 3)+D E^{2} &=6^{2} \\27+D E^{2} &=36 \\D E^{2} &=36-27 \\D E^{2} &=9 \\D E &=3\end{aligned}$

c. Use the fact that the triangles are similar to find the missing sides of $\triangle A B C$.

$\triangle A B C \sim \triangle D E F$, which means that the corresponding sides are proportional. Therefore,

$\begin{aligned}&\frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F} \\&\frac{A C}{D F}=\frac{18}{6}=3, \text { so the scale factor is } 3\end{aligned}$

• $\frac{A B}{D E}=3 \rightarrow \frac{A B}{3}=3 \rightarrow A B=9$
• $\frac{B C}{E F}=3 \rightarrow \frac{B C}{3 \sqrt{3}}=3 \rightarrow B C=9 \sqrt{3}$

Triangles of this configuration are called $30-60-90$ triangles because of their angle measures.

d. Explain why all 30-60-90 triangles are similar.

All 30-60-90 triangles are similar by $A A \sim$ because they all have at least two pairs of congruent angles.

1. Find the ratios between the three sides of any $30-60-90$ triangle.

[Figure 4]

The three sides of any 30-60-90 triangle will be in this ratio: $1: \sqrt{3}: 2$.

2. Find the missing sides of the triangle below.

[Figure 5]

The side opposite the $30^{\circ}$ angle is the smallest side because $30^{\circ}$ is the smallest angle. Therefore, the length of $10$ corresponds to the length of $1$ in the ratio $1: \sqrt{3}: 2$. The scale factor is $10$. The other sides of the triangle will be $10 \sqrt{3}$ and $20$, because $10: 10 \sqrt{3}: 20$ is equivalent to $1: \sqrt{3}: 2$. $B C=10 \sqrt{3}$ and $A C=20$.

Create similar triangles in order to solve for $x$.

[Figure 6]

[Figure 7]

$\triangle D G C \sim \triangle E G F \sim \triangle A G B$ by $A A \sim$ because angles $\angle D C G, \angle E F G, \angle A B G$ are all right angles and are therefore congruent and all triangles share $\angle G$. This means that their corresponding sides are proportional. First, solve for $G C$ by looking at $\triangle D G C$ and $\triangle E G F . \triangle D G C \sim \triangle E G F$ which means that corresponding sides are proportional.

Therefore:

$\begin{aligned} \frac{D G}{E G} &=\frac{G C}{G F}=\frac{D C}{E F} \\ \frac{D C}{G C} &=\frac{E F}{G F} \\ \frac{2}{G C} &=\frac{3.5}{2.5+G C} \\ 5+2 G C &=3.5 G C \\ 5 &=1.5 G C \\ G C & \approx 3.33 \end{aligned}$

Next, solve for $x$ by looking at $\triangle D G C$ and $\triangle A G B . \triangle D G C \sim \triangle A G B$ which means that corresponding sides are proportional.

Therefore:

$\begin{aligned} \frac{D G}{A G} &=\frac{G C}{G B}=\frac{D C}{A B} \\ \frac{D C}{G C} &=\frac{A B}{G B} \\ \frac{2}{3.33} &=\frac{x}{1.5+2.5+3.33} \\ \frac{2}{3.33} &=\frac{x}{7.33} \\ x &=4.4 \end{aligned}$

Example 1

Michael is 6 feet tall and is standing outside next to his younger sister. He notices that he can see both of their shadows and decides to measure each shadow. His shadow is 8 feet long and his sister's shadow is 5 feet long. How tall is Michael's sister?

You can answer this question by applying what you know about similar triangles.

The sun creates shadows at the same angle for both Michael and his sister. Assuming they are both standing up straight and making right angles with the ground, similar triangles are created.

[Figure 7]

Corresponding sides are proportional because the triangles are similar.

$\begin{aligned}\frac{\text { Michael's Height }}{\text { Sister's Height }} &=\frac{\text { Length of Michael's Shadow }}{\text { Length of Sister's Shadow }} \\\frac{6 ft }{\text { Sister's Height }} &=\frac{8 ft }{5 ft } \\\text { Sister's Height } &=\frac{6 \cdot 5}{8} \\&=3.75 ft\end{aligned}$

His sister is $3.75$ feet tall.

Example 2

Prove that all isosceles right triangles are similar.

Example 3

Find the measures of the angles of an isosceles right triangle. Why are isosceles right triangles called $45-45-90$ triangles?

• An isosceles right triangle is called a $45-45-90$ triangle because those are its angle measures.

Example 4

Use the Pythagorean Theorem to find the missing side of an isosceles right triangle whose legs are each length $x$.

[Figure 9]

The missing side is the hypotenuse of the right triangle, labeled $c$.

By the Pythagorean Theorem,

$\begin{aligned}A B^{2} &=A C^{2}+B C^{2} \\c^{2} &=x^{2}+x^{2} \\c^{2} &=2 x^{2} \\c &=\sqrt{2 x^{2}} \\c &=x \sqrt{2}\end{aligned}$

• The ratio of the sides of any isosceles right triangle will be $x: x: x \sqrt{2}$ which simplifies to $1: 1: \sqrt{2}$.

Example 5

Find the missing sides of the right triangle below without using the Pythagorean Theorem.

[Figure 10]