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}\\).
c no, because \\(\overline{ab}\\) is longer than \\(\overline{cd}\\).
d no, because \\(\overline{bc}\\) is not parallel to \\(\overline{ad}\\).

Explanation:

Step1: Recall slope formula

The slope of a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Calculate slope of \( \overline{AB} \)

For \(A(-3,0)\) and \(B(7,2)\), the slope \(m_{AB}=\frac{2 - 0}{7 - (-3)}=\frac{2}{10}=\frac{1}{5}\).

Step3: Calculate slope of \( \overline{CD} \)

For \(C(1,-4)\) and \(D(-9,-6)\), the slope \(m_{CD}=\frac{-6 - (-4)}{-9 - 1}=\frac{-2}{-10}=\frac{1}{5}\). So, \(m_{AB}=m_{CD}\), which means \( \overline{AB} \parallel \overline{CD} \).

Step4: Calculate slope of \( \overline{BC} \)

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

Step5: Calculate slope of \( \overline{AD} \)

For \(A(-3,0)\) and \(D(-9,-6)\), the slope \(m_{AD}=\frac{-6 - 0}{-9 - (-3)}=\frac{-6}{-6}=1\). So, \(m_{BC}=m_{AD}\), which means \( \overline{BC} \parallel \overline{AD} \).

Since both pairs of opposite sides are parallel, \(ABCD\) is a parallelogram. Option B states this reason.

Answer:

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