Question

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

Explanation:

Step1: Use the mid - point theorem

Since $\frac{CD}{CA}=\frac{CE}{CB}=\frac{13}{13 + 13}=\frac{11}{11+11}=\frac{1}{2}$, $DE\parallel AB$. By the mid - point theorem for similar triangles, if two triangles are similar and the ratio of the corresponding side lengths of the smaller triangle to the larger triangle is $k$, and the lengths of the corresponding parallel sides are related by the same ratio. Here, the ratio of the sides of $\triangle CDE$ to $\triangle CAB$ is $\frac{1}{2}$.

Step2: Calculate the length of $DE$

We know that $AB = 20$. Since $\triangle CDE\sim\triangle CAB$ and the ratio of their side lengths is $\frac{1}{2}$, we have $x=\frac{1}{2}\times AB$. Substituting $AB = 20$ into the formula, we get $x = 10$.

Answer:

$10$