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A, B, C, and D have the coordinates [tex](-8,1)[/tex], [tex](-2,4)[/tex], [tex](-3,-1)[/tex], and [tex](-6,5)[/tex], respectively. Which sentence about the points is true?

A. A, B, C, and D lie on the same line.
B. [tex]\overleftrightarrow{AB}[/tex] and [tex]\overleftrightarrow{CD}[/tex] are perpendicular lines.
C. [tex]\overleftrightarrow{AB}[/tex] and [tex]\overleftrightarrow{CD}[/tex] are parallel lines.
D. [tex]\overleftrightarrow{AB}[/tex] and [tex]\overleftrightarrow{CD}[/tex] are intersecting lines but are not perpendicular.
E. [tex]\overleftrightarrow{AC}[/tex] and [tex]\overleftrightarrow{BD}[/tex] are parallel lines.



Answer :

GinnyAnswer
To determine which sentence about the points is true, we need to analyze the slopes of the lines formed by these points. Let's start by considering the lines [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex].

1. Calculate the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{-2 - (-8)} = \frac{3}{6} = 0.5 \][/tex]

2. Calculate the slope of line [tex]\(\overleftrightarrow{CD}\)[/tex]:
[tex]\[ \text{slope}_{CD} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{-6 - (-3)} = \frac{6}{-3} = -2.0 \][/tex]

3. Determine if the lines are parallel:
Lines are parallel if their slopes are equal. Here, [tex]\(\text{slope}_{AB} = 0.5\)[/tex] and [tex]\(\text{slope}_{CD} = -2.0\)[/tex]. Since [tex]\(0.5 \ne -2.0\)[/tex], [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are not parallel.

4. Determine if the lines are perpendicular:
Lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. We calculate:
[tex]\[ \text{slope}_{AB} \times \text{slope}_{CD} = 0.5 \times (-2.0) = -1.0 \][/tex]
Since the product indeed equals [tex]\(-1.0\)[/tex], [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are perpendicular.

Conclusively, the correct answer is:

B. [tex]\( \overleftrightarrow{AB}\)[/tex] and [tex]\( \overleftrightarrow{CD}\)[/tex] are perpendicular lines.

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