2. pqrs is a trapezoid, then what is the length of the midsegment. p(-3,3) q(3,5) r(5,-1) s(-7,-5)

Question

Accepted Answer

# Explanation:
## Step1: Find the length of the two parallel sides (bases)
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $PQ$ with $P(-3,3)$ and $Q(3,5)$:
$$
\begin{align*}
d_{PQ}&=\sqrt{(3-(-3))^2+(5 - 3)^2}\
&=\sqrt{(6)^2+(2)^2}\
&=\sqrt{36 + 4}\
&=\sqrt{40}=2\sqrt{10}
\end{align*}
$$
For side $SR$ with $S(-7,-5)$ and $R(5,-1)$:
$$
\begin{align*}
d_{SR}&=\sqrt{(5-(-7))^2+(-1-(-5))^2}\
&=\sqrt{(12)^2+(4)^2}\
&=\sqrt{144+16}\
&=\sqrt{160}=4\sqrt{10}
\end{align*}
$$

## Step2: Use the mid - segment formula
The formula for the length of the mid - segment $m$ of a trapezoid is $m=\frac{b_1 + b_2}{2}$, where $b_1$ and $b_2$ are the lengths of the two parallel sides.
Here $b_1 = 2\sqrt{10}$, $b_2=4\sqrt{10}$
$$
\begin{align*}
m&=\frac{2\sqrt{10}+4\sqrt{10}}{2}\
&=\frac{6\sqrt{10}}{2}\
&=3\sqrt{10}
\end{align*}
$$

# Answer:
$3\sqrt{10}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the length of the two parallel sides (bases)

Step2: Use the mid - segment formula

Answer: