Exercise 15 Page 559

We are given following diagram and want to find the length of the log. We will identify the vertices of the triangles with letters to make our calculations easier. Let's first show that triangles $XTZ$ and $YWZ$ are similar. Once we know that the triangles are similar, we can use their proportional relationship to find the missing distance.

For $△XTZ$ and $△YWZ$ to be similar, at least two of their angles need to be congruent. We can see from the angle markers shown in the diagram that $\angle X$ is congruent to $\angle Y$ and that $\angle T$ is congruent to $\angle W.$ Since two angles of $△XTZ$ are congruent to two angles of $△YWZ,$ they are similar by the Angle-Angle (AA) Similarity rule. Now, notice that $\overline{XT}$ corresponds to $\overline{YW}$ and that $\overline{XZ}$ corresponds to $\overline{YZ}.$ Corresponding sides of similar figures will have proportional lengths. We know that $\overline{XT}=8$ meters, $\overline{WZ}=4$ meters, and $\overline{TZ}=12$ meters, so we will use these lengths to write a proportion. Let's solve this proportion using cross products. The log is $6$ meters long.