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 $\angle CDE=\angle CAB$, $\angle CED=\angle CBA$ (corresponding angles for parallel lines $DE\parallel AB$), $\triangle CDE\sim\triangle CAB$.

Step2: Set up the proportion

The ratio of corresponding sides of similar triangles is equal. We have $\frac{CD}{CA}=\frac{DE}{AB}$. Given $CD = 12$, $CA=12 + 12=24$, and $DE = 12$. Let $AB=x$. Then the proportion is $\frac{12}{24}=\frac{12}{x}$.

Step3: Solve the proportion

Cross - multiply: $12x=12\times24$. Then $x = 24$.

Answer:

$24$