Question

written response
as shown in the figure, in △abc, d and e are on $overline{ac}$ and $overline{ab}$, respectively.
$mangle aed = mangle c$, $ab = 10$, $ad = 6$, and $ac = 8$. what is the length of $overline{ae}$?

Explanation:

Step1: Prove triangle similarity

Since $\angle AED=\angle C$ and $\angle A=\angle A$ (common - angle), by the AA (angle - angle) similarity criterion, $\triangle AED\sim\triangle ACB$.

Step2: Set up the proportion

For similar triangles $\triangle AED$ and $\triangle ACB$, the ratios of corresponding sides are equal. That is, $\frac{AE}{AC}=\frac{AD}{AB}$.

Step3: Substitute the given values

We know that $AB = 10$, $AD = 6$, and $AC = 8$. Substituting these values into the proportion $\frac{AE}{8}=\frac{6}{10}$.

Step4: Solve for $AE$

Cross - multiply to get $10\times AE=6\times8$, so $10AE = 48$. Then $AE=\frac{48}{10}=4.8$.

Answer:

$4.8$