Exercise 6 Page 557

We are given following diagram and want to find the height of the brace. Let's add some letters to our diagram to make it easier to reference the different parts of the triangles. We will first show that triangles $ABC$ and $ADE$ are similar. Once we know that the triangles are similar, we can use their proportional relationship to find the missing distance.

From the angle markers shown in the diagram, we can see that $\angle B$ is congruent to $\angle D$ and that $\angle C$ is congruent to $\angle E.$ Since two angles of $△ABC$ are congruent to two angles of $△ADE,$ they are similar by the Angle-Angle (AA) Similarity rule. Now, notice that $\overline{AC}$ corresponds to $\overline{AE}$ and $\overline{BC}$ corresponds to $\overline{DE}.$ Remember that corresponding sides of similar figures will have proportional lengths. We know that $\overline{AC}=7$ feet, $\overline{AE}=15$ feet, and $\overline{DE}=9$ feet. We can use these lengths to write a proportion. Let's solve this proportion using cross products. The height of the brace is $4.2$ feet.