Question

quadrilateral abcd has the following vertices:

• a(-3,0)
• b(7,2)
• c(1,-4)
• d(-9,-6)

is quadrilateral abcd a parallelogram, and why?
choose 1 answer:
a yes, because $overline{ab} parallel overline{cd}$, and $overline{bc} perp overline{ab}$.
b yes, because $overline{ab} parallel overline{cd}$, and $overline{bc} parallel overline{ad}$

Explanation:

Step1: Calculate the slope of $\overline{AB}$

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-3,0)$ and $B(7,2)$, $m_{AB}=\frac{2 - 0}{7-(-3)}=\frac{2}{10}=\frac{1}{5}$.

Step2: Calculate the slope of $\overline{CD}$

For points $C(1,-4)$ and $D(-9,-6)$, $m_{CD}=\frac{-6 - (-4)}{-9 - 1}=\frac{-2}{-10}=\frac{1}{5}$.

Step3: Calculate the slope of $\overline{BC}$

For points $B(7,2)$ and $C(1,-4)$, $m_{BC}=\frac{-4 - 2}{1 - 7}=\frac{-6}{-6}=1$.

Step4: Calculate the slope of $\overline{AD}$

For points $A(-3,0)$ and $D(-9,-6)$, $m_{AD}=\frac{-6 - 0}{-9-(-3)}=\frac{-6}{-6}=1$.

Step5: Check parallel - side condition

Since $m_{AB}=m_{CD}=\frac{1}{5}$, so $\overline{AB}\parallel\overline{CD}$. Since $m_{BC}=m_{AD} = 1$, so $\overline{BC}\parallel\overline{AD}$. A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Answer:

B. Yes, because $\overline{AB}\parallel\overline{CD}$, and $\overline{BC}\parallel\overline{AD}$