Question

here are the endpoints of the segments $overline{pq}$, $overline{wx}$, and $overline{yz}$.
$p(-4,5), q(-7,7)$
$w(1, - 6), x(-1,-3)$
$y(3,0), z(5,3)$
follow the directions below.
(a) find the length of each segment. give an exact answer (not a decimal approximation).
$pq=square$
$wx=square$
$yz=square$
(b) check all statements that are true below.
$squareoverline{pq}congoverline{wx}$
$squareoverline{pq}congoverline{yz}$
$squareoverline{wx}congoverline{yz}$
$square$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 $P(-4,5)$ and $Q(-7,7)$, we have $x_1=-4,y_1 = 5,x_2=-7,y_2 = 7$. Then $PQ=\sqrt{(-7+4)^2+(7 - 5)^2}=\sqrt{(-3)^2+2^2}=\sqrt{9 + 4}=\sqrt{13}$.

Step3: Calculate length of $WX$

For $W(1,-6)$ and $X(-1,-3)$, we have $x_1 = 1,y_1=-6,x_2=-1,y_2=-3$. Then $WX=\sqrt{(-1 - 1)^2+(-3 + 6)^2}=\sqrt{(-2)^2+3^2}=\sqrt{4+9}=\sqrt{13}$.

Step4: Calculate length of $YZ$

For $Y(3,0)$ and $Z(5,3)$, we have $x_1 = 3,y_1 = 0,x_2=5,y_2 = 3$. Then $YZ=\sqrt{(5 - 3)^2+(3 - 0)^2}=\sqrt{2^2+3^2}=\sqrt{4 + 9}=\sqrt{13}$.

Step5: Check congruence

Since $PQ=\sqrt{13}$, $WX=\sqrt{13}$ and $YZ=\sqrt{13}$, we have $\overline{PQ}\cong\overline{WX}$, $\overline{PQ}\cong\overline{YZ}$ and $\overline{WX}\cong\overline{YZ}$.

Answer:

(a)
$PQ=\sqrt{13}$
$WX=\sqrt{13}$
$YZ=\sqrt{13}$
(b)
$\overline{PQ}\cong\overline{WX}$
$\overline{PQ}\cong\overline{YZ}$
$\overline{WX}\cong\overline{YZ}$