Question

here are the endpoints of the segments $overline{pq}$, $overline{rs}$, and $overline{tv}$.
$p(-6,8)$, $q(2,7)$
$r(1, - 1)$, $s(3,-8)$
$t(-2,1)$, $v(-1,-7)$
follow the directions below.
(a) find the length of each segment. give an exact answer (not a decimal approximation).
$pq=square$
$rs=square$
$tv=square$
(b) check all statements that are true below.
$squareoverline{pq}congoverline{rs}$
$squareoverline{pq}congoverline{tv}$
$squareoverline{rs}congoverline{tv}$
$square$ none of these are true

Explanation:

Step1: Recall 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 length of $PQ$

For $P(-6,8)$ and $Q(2,7)$, we have $x_1=-6,y_1 = 8,x_2=2,y_2 = 7$. Then $PQ=\sqrt{(2-(-6))^2+(7 - 8)^2}=\sqrt{(2 + 6)^2+(-1)^2}=\sqrt{64 + 1}=\sqrt{65}$.

Step3: Calculate length of $RS$

For $R(1,-1)$ and $S(3,-8)$, we have $x_1 = 1,y_1=-1,x_2=3,y_2=-8$. Then $RS=\sqrt{(3 - 1)^2+(-8-(-1))^2}=\sqrt{2^2+(-7)^2}=\sqrt{4 + 49}=\sqrt{53}$.

Step4: Calculate length of $TV$

For $T(-2,1)$ and $V(-1,-7)$, we have $x_1=-2,y_1 = 1,x_2=-1,y_2=-7$. Then $TV=\sqrt{(-1-(-2))^2+(-7 - 1)^2}=\sqrt{(-1 + 2)^2+(-8)^2}=\sqrt{1+64}=\sqrt{65}$.

Step5: Check congruence

Since $PQ=\sqrt{65}$ and $TV=\sqrt{65}$, $PQ\cong TV$.

Answer:

(a)
$PQ=\sqrt{65}$
$RS=\sqrt{53}$
$TV=\sqrt{65}$
(b)
$\overline{PQ}\cong\overline{TV}$