use the distance formula to find the lengths of the sides ($ \overline { a c } $ and $ \overline { a e } $) that include the overlapping angle. the length of $ \overline { a c } $ is $ \sqrt { 2 } $. the length of $ \overline { a e } $ is $ \sqrt { 2 } $.

Question

Accepted Answer

# Explanation:
## Step1: Recall the distance formula
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
## Step2: Calculate the length of $\overline{AC}$
For points $A(3,1)$ and $C(5,3)$, $x_1 = 3,y_1 = 1,x_2 = 5,y_2 = 3$.
$$
\begin{align*}
AC&=\sqrt{(5 - 3)^2+(3 - 1)^2}\
&=\sqrt{2^2+2^2}\
&=\sqrt{4 + 4}\
&=\sqrt{8}\
&=2\sqrt{2}
\end{align*}
$$
## Step3: Calculate the length of $\overline{AE}$
For points $A(3,1)$ and $E(7,5)$, $x_1 = 3,y_1 = 1,x_2 = 7,y_2 = 5$.
$$
\begin{align*}
AE&=\sqrt{(7 - 3)^2+(5 - 1)^2}\
&=\sqrt{4^2+4^2}\
&=\sqrt{16+16}\
&=\sqrt{32}\
&=4\sqrt{2}
\end{align*}
$$

# Answer:
The length of $\overline{AC}$ is $2\sqrt{2}$. The length of $\overline{AE}$ is $4\sqrt{2}$.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall the distance formula

Step2: Calculate the length of \(\overline{AC}\)

Step3: Calculate the length of \(\overline{AE}\)

Answer: