Exercise 42 Page 820

Savannah and Leon are standing $8$ feet apart in front of a rock climbing wall. We want to find the height of the wall, $h.$

First, notice that the bigger triangle is a $45^{\circ }\text{-}45^{\circ }\text{-}90^{\circ }$ triangle. In this type of triangle the legs are congruent. Therefore, the distance between Savannah and the bottom of the climbing wall equals the height of the wall. Then, the distance between Leon and the wall will be a difference between height and the distance between Savannah and Leon.

Now, let's consider the smaller right triangle. It has a marked angle measuring $60^{\circ }.$ By the Triangle Angle Sum Theorem, the measure of the third angle must be $30^{\circ }.$ Therefore, it is a $30^{\circ }\text{-}60^{\circ }\text{-}90^{\circ }$ triangle. Let's take a closer look at it.

We know the measures of two angles and the expressions for the sides opposite to them. We can use the Law of Sines to form an equation in terms of $h.$

Let's substitute $b=h,$ $c=h-8,$ $m\angle B=60^{\circ },$ and $m\angle C=30^{\circ }$ to solve for $h.$ The height of the rock climbing wall is about $18.9$ feet.