Exercise 24 Page 624

Let's begin with recalling the Geometric Mean Altitude Theorem. Now, let's take a look at the given picture. We will label the vertices with consecutive letters. Let $x$ represents the height of the gym ceiling above Evelina's line of vision. As we can see, $\overline{BD}$ is an altitude of a right triangle $ABC.$ Therefore, we can write that the length of $\overline{BD}$ is the geometric mean between the lengths of $\overline{AD}$ and $\overline{DC}.$ Next, we will solve the above equation for $x$ using inverse operations. The length of $\overline{AD}$ is $20$ feet. With this information, we can evaluate the height of the gym ceiling. To do this, we will add the lengths of $\overline{AD}$ and $\overline{DC}.$ The height of the gym ceiling is $25$ feet. Since Evelina wants the ends of the strings where the stars will be attached to be $7$ feet from the floor, we need to subtract $7$ from the height of the gym ceiling to evaluate the strings length. She should make the strings of the length of $18$ feet.