here are the endpoints of the segments \overline{pq}, \overline{rs}, and \overline{tv}.
p(3,1), q(-2,-5)
r(-8,0), s(-4,-7)
t(-7,8), v(-2,2)
follow the directions below.
(a) find the length of each segment.
give an exact answer (not a decimal approximation).
pq =
rs =
tv =

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 $PQ$
For $P(3,1)$ and $Q(-2,-5)$, $x_1 = 3,y_1 = 1,x_2=-2,y_2=-5$.
$$
\begin{align*}
PQ&=\sqrt{(-2 - 3)^2+(-5 - 1)^2}\
&=\sqrt{(-5)^2+(-6)^2}\
&=\sqrt{25 + 36}\
&=\sqrt{61}
\end{align*}
$$
## Step3: Calculate the length of $RS$
For $R(-8,0)$ and $S(-4,-7)$, $x_1=-8,y_1 = 0,x_2=-4,y_2=-7$.
$$
\begin{align*}
RS&=\sqrt{(-4+8)^2+(-7 - 0)^2}\
&=\sqrt{(4)^2+(-7)^2}\
&=\sqrt{16 + 49}\
&=\sqrt{65}
\end{align*}
$$
## Step4: Calculate the length of $TV$
For $T(-7,8)$ and $V(-2,2)$, $x_1=-7,y_1 = 8,x_2=-2,y_2=2$.
$$
\begin{align*}
TV&=\sqrt{(-2 + 7)^2+(2 - 8)^2}\
&=\sqrt{(5)^2+(-6)^2}\
&=\sqrt{25+36}\
&=\sqrt{61}
\end{align*}
$$

# Answer:
$PQ=\sqrt{61}$, $RS=\sqrt{65}$, $TV=\sqrt{61}$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the distance formula

Step2: Calculate the length of \(PQ\)

Step3: Calculate the length of \(RS\)

Step4: Calculate the length of \(TV\)

Answer: