Question

find x, the length of the line segment $overline{ab}$.

Explanation:

Step1: Use similarity of triangles

Since $\triangle CDE$ and $\triangle CAB$ are similar (by AA - similarity as $\angle C$ is common and the ratios of the sides are in proportion). The ratio of the corresponding sides of similar triangles is equal. The ratio of the sides of $\triangle CDE$ to $\triangle CAB$ for the sides along $CA$ and $CB$ is $\frac{CD}{CA}=\frac{14}{14 + 14}=\frac{14}{28}=\frac{1}{2}$ and $\frac{CE}{CB}=\frac{15}{15 + 15}=\frac{15}{30}=\frac{1}{2}$.

Step2: Set up proportion for side lengths

For similar triangles $\triangle CDE$ and $\triangle CAB$, we have the proportion $\frac{DE}{AB}=\frac{CD}{CA}$. We know that $DE = 17$, $CD=14$, and $CA = 28$. Substituting the values into the proportion $\frac{17}{x}=\frac{14}{28}$.

Step3: Solve the proportion for $x$

Cross - multiply: $14x=17\times28$. Then $x=\frac{17\times28}{14}$. Simplifying, $\frac{17\times28}{14}=17\times2 = 34$.

Answer:

$34$