Question

justify your answer using the indicated formula.

1. p(-3, -2), q(-4, 2), r(2, 4), s(3, 0);

slope formula

1. j(-6, 3), k(0, 6), l(2, 2), m(-4, -1);

distance formula

Explanation:

Problem 23:

Step1: Define slope formula

Slope between $(x_1,y_1)$ & $(x_2,y_2)$: $m=\frac{y_2-y_1}{x_2-x_1}$

Step2: Calculate slope of $PQ$

Points $P(-3,-2), Q(-4,2)$
$m_{PQ}=\frac{2-(-2)}{-4-(-3)}=\frac{4}{-1}=-4$

Step3: Calculate slope of $RS$

Points $R(2,4), S(3,0)$
$m_{RS}=\frac{0-4}{3-2}=\frac{-4}{1}=-4$

Step4: Calculate slope of $QR$

Points $Q(-4,2), R(2,4)$
$m_{QR}=\frac{4-2}{2-(-4)}=\frac{2}{6}=\frac{1}{3}$

Step5: Calculate slope of $SP$

Points $S(3,0), P(-3,-2)$
$m_{SP}=\frac{-2-0}{-3-3}=\frac{-2}{-6}=\frac{1}{3}$

Problem 24:

Step1: Define distance formula

Distance between $(x_1,y_1)$ & $(x_2,y_2)$: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Step2: Calculate length of $JK$

Points $J(-6,3), K(0,6)$
$d_{JK}=\sqrt{(0-(-6))^2+(6-3)^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$

Step3: Calculate length of $LM$

Points $L(2,2), M(-4,-1)$
$d_{LM}=\sqrt{(-4-2)^2+(-1-2)^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$

Step4: Calculate length of $KL$

Points $K(0,6), L(2,2)$
$d_{KL}=\sqrt{(2-0)^2+(2-6)^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt{5}$

Step5: Calculate length of $MJ$

Points $M(-4,-1), J(-6,3)$
$d_{MJ}=\sqrt{(-6-(-4))^2+(3-(-1))^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt{5}$

Answer:

Problem 23:

Since $m_{PQ}=m_{RS}$ and $m_{QR}=m_{SP}$, opposite sides are parallel, so $PQRS$ is a parallelogram.

Problem 24:

Since $d_{JK}=d_{LM}$ and $d_{KL}=d_{MJ}$, opposite sides are congruent, so $JKLM$ is a parallelogram.