## Application of Similar Triangles

Read this section and watch the videos to see the examples of applications of similar triangles in geometric and real-life problems.

### Examples

#### 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.

Consider two generic isosceles right triangles:

Two pairs of sides are proportional with a ratio of $\frac{b}{a}$. Also, $\angle C \cong \angle F$. Therefore, the two triangles are similar by $S A S \sim$.

#### Example 3

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

The base angles of an isosceles triangle are congruent. If the vertex angle is $90^{\circ}$, each base angle is $\frac{180^{\circ}-90^{\circ}}{2}=45^{\circ}$. The measures of the angles of an isosceles right triangle are $45$, $45$, and $90$.

• 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]

If one of the legs is $3$, then the other leg is also $3$, so $A C=3$. The ratio of the sides of an isosceles right triangle is $1: 1: \sqrt{2}$. Therefore, the hypotenuse will be $3 \sqrt{2}$, so $A B=3 \sqrt{2}$.