Exercise 14 Page 559

We are given following diagram and we want to find the distance across the river. Let's first show that triangles $CBA$ and $DBE$ are similar. Once we know that the triangles are similar, we can use their proportional relationship to find the missing distance.

From the picture, we can see that $\angle A$ and $\angle E$ are right angles. This means that they have the same measures. They are congruent. Also we can see that $\angle CBA$ and $\angle DBE$ are vertical angles. We know that vertical angles are congruent. Let's mark it on our graph.

Since two angles of $△CBA$ are congruent to two angles of $△DBE,$ they are similar by the rule for Angle-Angle (AA) Similarity. Now, notice that $\overline{BE}$ corresponds to $\overline{BA}$ and $\overline{AC}$ corresponds to $\overline{ED}.$ Remember that corresponding sides of similar figures will have proportional lengths. We know that $BE=350$ meters, $BA=140$ meters, and $ED=300$ meters. We can use these lengths to write a proportion. Let's solve this proportion using cross products. The distance across the river is $120$ meters.