Question

the length of $overline{ab}$ is 20 in the figure shown. what is the length of $overline{ae}$?

Explanation:

Step1: Identify the right - triangle relationship

In right - triangle $ABE$, $\angle ABE = 30^{\circ}$, and the hypotenuse $AB = 20$.

Step2: Apply the cosine function

The cosine of an angle in a right - triangle is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$, and we want to find the length of the adjacent side $AE$ to $\angle ABE$. So, $\cos30^{\circ}=\frac{AE}{AB}$.
Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$ and $AB = 20$, we have $AE = AB\times\cos30^{\circ}$.

Step3: Calculate the length of $AE$

Substitute $AB = 20$ into the formula: $AE=20\times\frac{\sqrt{3}}{2}=10\sqrt{3}$.

Answer:

$10\sqrt{3}$