Question

is the quadrilateral a parallelogram? a(-4,1) b(-1,3) c(3,1) d(0,-1) a. yes. all opposite sides are parallel and congruent b. no. the opposite sides are not parallel and not congruent

Explanation:

Step1: Calculate slope of AB

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-4,1)$ and $B(-1,3)$, $m_{AB}=\frac{3 - 1}{-1-(-4)}=\frac{2}{3}$.

Step2: Calculate slope of CD

For points $C(3,1)$ and $D(0,-1)$, $m_{CD}=\frac{-1 - 1}{0 - 3}=\frac{-2}{-3}=\frac{2}{3}$.

Step3: Calculate slope of BC

For points $B(-1,3)$ and $C(3,1)$, $m_{BC}=\frac{1 - 3}{3-(-1)}=\frac{-2}{4}=-\frac{1}{2}$.

Step4: Calculate slope of DA

For points $D(0,-1)$ and $A(-4,1)$, $m_{DA}=\frac{1-(-1)}{-4 - 0}=\frac{2}{-4}=-\frac{1}{2}$.

Step5: Check parallel - side condition

Since $m_{AB}=m_{CD}=\frac{2}{3}$ and $m_{BC}=m_{DA}=-\frac{1}{2}$, opposite sides are parallel.

Step6: Calculate length of AB

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $A(-4,1)$ and $B(-1,3)$, $d_{AB}=\sqrt{(-1 + 4)^2+(3 - 1)^2}=\sqrt{9 + 4}=\sqrt{13}$.

Step7: Calculate length of CD

For $C(3,1)$ and $D(0,-1)$, $d_{CD}=\sqrt{(0 - 3)^2+(-1 - 1)^2}=\sqrt{9+4}=\sqrt{13}$.

Step8: Calculate length of BC

For $B(-1,3)$ and $C(3,1)$, $d_{BC}=\sqrt{(3 + 1)^2+(1 - 3)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$.

Step9: Calculate length of DA

For $D(0,-1)$ and $A(-4,1)$, $d_{DA}=\sqrt{(-4 - 0)^2+(1+1)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$.

Step10: Check congruent - side condition

Since $d_{AB}=d_{CD}$ and $d_{BC}=d_{DA}$, opposite sides are congruent.

Answer:

A. Yes. All opposite sides are parallel and congruent