Question

segments and angles identifying congruent segments in the plane here are the endpoints of the segments $overline{pq}$, $overline{rs}$, and $overline{tv}$. $p(7,4), q(2,8)$ $r(5,0), s(1, - 5)$ $t(-3,5), v(-8,1)$ 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 $P(7,4)$ and $Q(2,8)$, $x_1 = 7,y_1 = 4,x_2=2,y_2 = 8$. Then $PQ=\sqrt{(2 - 7)^2+(8 - 4)^2}=\sqrt{(-5)^2+4^2}=\sqrt{25 + 16}=\sqrt{41}$.

Step3: Calculate length of $RS$

For $R(5,0)$ and $S(1,-5)$, $x_1 = 5,y_1 = 0,x_2=1,y_2=-5$. Then $RS=\sqrt{(1 - 5)^2+(-5 - 0)^2}=\sqrt{(-4)^2+(-5)^2}=\sqrt{16 + 25}=\sqrt{41}$.

Step4: Calculate length of $TV$

For $T(-3,5)$ and $V(-8,1)$, $x_1=-3,y_1 = 5,x_2=-8,y_2 = 1$. Then $TV=\sqrt{(-8+3)^2+(1 - 5)^2}=\sqrt{(-5)^2+(-4)^2}=\sqrt{25 + 16}=\sqrt{41}$.

Step5: Check congruence

Since $PQ=\sqrt{41}$, $RS=\sqrt{41}$, and $TV=\sqrt{41}$, we have $\overline{PQ}\cong\overline{RS}$, $\overline{PQ}\cong\overline{TV}$, $\overline{RS}\cong\overline{TV}$.

Answer:

(a) $PQ=\sqrt{41}$
$RS=\sqrt{41}$
$TV=\sqrt{41}$
(b) $\overline{PQ}\cong\overline{RS}$
$\overline{PQ}\cong\overline{TV}$
$\overline{RS}\cong\overline{TV}$