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



Answer :

GinnyAnswer
Let's analyze the problem step-by-step.

1. Identify the coordinates:
- [tex]\( A(-8, 1) \)[/tex]
- [tex]\( B(-2, 4) \)[/tex]
- [tex]\( C(-3, -1) \)[/tex]
- [tex]\( D(-6, 5) \)[/tex]

2. Calculate the slopes of line segments [tex]\( \overrightarrow{AB} \)[/tex] and [tex]\( \overrightarrow{CD} \)[/tex]:

- Slope of [tex]\( \overrightarrow{AB} \)[/tex]:
[tex]\[ \text{Slope of } \overrightarrow{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{-2 - (-8)} = \frac{3}{6} = \frac{1}{2} \][/tex]

- Slope of [tex]\( \overrightarrow{CD} \)[/tex]:
[tex]\[ \text{Slope of } \overrightarrow{CD} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{-6 - (-3)} = \frac{5 + 1}{-6 + 3} = \frac{6}{-3} = -2 \][/tex]

3. Determine the relationship between the two slopes:

- Parallel lines:
- Two lines are parallel if their slopes are equal. Here, the slopes of [tex]\( \overrightarrow{AB} \)[/tex] and [tex]\( \overrightarrow{CD} \)[/tex] are [tex]\(\frac{1}{2}\)[/tex] and [tex]\(-2\)[/tex], respectively. Since [tex]\(\frac{1}{2} \neq -2\)[/tex], the lines are not parallel.

- Perpendicular lines:
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Calculate the product of the slopes:
[tex]\[ \left(\frac{1}{2}\right) \times (-2) = -1 \][/tex]
- Since the product of the slopes is [tex]\(-1\)[/tex], the lines [tex]\( \overrightarrow{AB} \)[/tex] and [tex]\( \overrightarrow{CD} \)[/tex] are perpendicular.

Given these steps, we can conclude that the true statement is:
[tex]\[ \text{B. } \overrightarrow{A B} \text{ and } \overrightarrow{C D} \text{ are perpendicular lines.} \][/tex]

Hence, the correct answer is [tex]\( \boxed{2} \)[/tex].

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