Question

a, b, c, and d have the coordinates (-8, 1), (-2, 4), (-3, -1), and (-6, 5), respectively. which sentence about the points is true?
a. a, b, c, and d lie on the same line.
b. $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ are perpendicular lines.
c. $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ are parallel lines.
d. $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ are intersecting lines but are not perpendicular.
e. $overleftrightarrow{ac}$ and $overleftrightarrow{bd}$ are parallel lines.

Explanation:

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

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-8,1)$ and $B(-2,4)$, $m_{AB}=\frac{4 - 1}{-2-(-8)}=\frac{3}{6}=\frac{1}{2}$.

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

For points $C(-3,-1)$ and $D(-6,5)$, $m_{CD}=\frac{5-(-1)}{-6 - (-3)}=\frac{6}{-3}=- 2$.

Step3: Analyze the relationship between the lines

Since $m_{AB}\times m_{CD}=\frac{1}{2}\times(-2)=-1$, $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are perpendicular lines.

Answer:

B. $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are perpendicular lines.