Question

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

Explanation:

Step1: Identify similar - triangles

Since $\triangle CDE$ and $\triangle CAB$ share the same - angle at $C$ and $DE\parallel AB$, $\triangle CDE\sim\triangle CAB$ by the AA (angle - angle) similarity criterion.

Step2: Set up the proportion

For similar triangles, the ratios of corresponding sides are equal. The ratio of the sides of $\triangle CDE$ to $\triangle CAB$ is given by $\frac{CD}{CA}=\frac{CE}{CB}=\frac{DE}{AB}$. We know that $CD = 8$, $CA=8 + 8=16$, $DE = 8$. Let $AB=x$. Then the proportion is $\frac{CD}{CA}=\frac{DE}{AB}$, which is $\frac{8}{16}=\frac{8}{x}$.

Step3: Solve the proportion for $x$

Cross - multiply the proportion $\frac{8}{16}=\frac{8}{x}$ to get $8x=8\times16$. Then $x = 16$.

Answer:

$16$