Question

segments and angles identifying congruent segments in the plane here are the endpoints of the segments $overline{pq}$, $overline{rs}$, and $overline{tv}$. $p(-6,3),q(-4,0)$ $r(-3,-4),s(0,-2)$ $t(4,2),v(5,-2)$ 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 $\overline{PQ}$

For $P(-6,3)$ and $Q(-4,0)$, we have $x_1=-6,y_1 = 3,x_2=-4,y_2 = 0$. Then $PQ=\sqrt{(-4-(-6))^2+(0 - 3)^2}=\sqrt{(2)^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}$.

Step3: Calculate length of $\overline{RS}$

For $R(-3,-4)$ and $S(0,-2)$, we have $x_1=-3,y_1=-4,x_2 = 0,y_2=-2$. Then $RS=\sqrt{(0-(-3))^2+(-2-(-4))^2}=\sqrt{(3)^2+(2)^2}=\sqrt{9+4}=\sqrt{13}$.

Step4: Calculate length of $\overline{TV}$

For $T(4,2)$ and $V(5,-2)$, we have $x_1 = 4,y_1=2,x_2=5,y_2=-2$. Then $TV=\sqrt{(5 - 4)^2+(-2 - 2)^2}=\sqrt{(1)^2+(-4)^2}=\sqrt{1 + 16}=\sqrt{17}$.

Step5: Check congruence

Since $PQ=\sqrt{13}$ and $RS=\sqrt{13}$, $\overline{PQ}\cong\overline{RS}$. Since $PQ
eq TV$ and $RS
eq TV$, the other congruence - statements are false.

Answer:

(a) $PQ=\sqrt{13}$, $RS=\sqrt{13}$, $TV=\sqrt{17}$
(b) $\overline{PQ}\cong\overline{RS}$