Question

in △acb shown below, point d is on (overline{bc}), and point e is on (overline{ac}) so that (angle abc cong angle dec). if (bc = 14), (ec = 6), and (ca = 21), find the length of (overline{cd}). figures are not necessarily drawn to scale.

triangle abc with point d on bc and point e on ac, and segment de connecting them. answer box for cd and submit answer button are present.

Explanation:

Step1: Identify similar triangles

We know $\angle ABC \cong \angle DEC$ and $\angle C$ is common to both $\triangle ACB$ and $\triangle DCE$. By AA (Angle-Angle) similarity criterion, $\triangle ACB \sim \triangle DCE$.

Step2: Set up proportion for sides

For similar triangles, corresponding sides are proportional:
$\frac{CD}{CA} = \frac{EC}{BC}$

Step3: Substitute given values

Substitute $BC=14$, $EC=6$, $CA=21$:
$\frac{CD}{21} = \frac{6}{14}$

Step4: Solve for $CD$

Simplify $\frac{6}{14}=\frac{3}{7}$, then solve:
$CD = 21 \times \frac{3}{7}$
$CD = 9$

Answer:

$9$