Question

graph each quadrilateral with the given vertices.
determine whether the figure is a rectangle.
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:

For Problem 23:

Step1: Define slope formula

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$

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 QR

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

Step4: Calculate slope of RS

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

Step5: Calculate slope of SP

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

Step6: Check perpendicularity

Perpendicular slopes multiply to $-1$.
$m_{PQ} \times m_{QR} = -4 \times \frac{1}{3}=-\frac{4}{3}
eq -1$
$m_{QR} \times m_{RS} = \frac{1}{3} \times -4=-\frac{4}{3}
eq -1$

Step1: Define distance formula

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

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 KL

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

Step4: Calculate length of LM

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

Step5: Calculate length of MJ

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

Step6: Calculate diagonal JL

Points $J(-6,3), L(2,2)$:
$d_{JL}=\sqrt{(2-(-6))^2+(2-3)^2}=\sqrt{64+1}=\sqrt{65}$

Step7: Calculate diagonal KM

Points $K(0,6), M(-4,-1)$:
$d_{KM}=\sqrt{(-4-0)^2+(-1-6)^2}=\sqrt{16+49}=\sqrt{65}$

Step8: Verify rectangle conditions

In a rectangle, opposite sides are equal, and diagonals are equal. Here, $d_{JK}=d_{LM}$, $d_{KL}=d_{MJ}$, and $d_{JL}=d_{KM}$.

Answer:

The quadrilateral PQRS is not a rectangle. Opposite sides are parallel (equal slopes), but adjacent sides are not perpendicular (their slopes do not multiply to -1).

---

For Problem 24: