Answer :
To determine which statement about the points [tex]\( A(-8,1) \)[/tex], [tex]\( B(-2,4) \)[/tex], [tex]\( C(-3,-1) \)[/tex], and [tex]\( D(-6,5) \)[/tex] is true, we need to analyze the slopes of the lines formed by these points. Below are the steps to find out the relationships between the lines [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex]:
1. Calculate the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
The slope formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates of points [tex]\( A(-8,1) \)[/tex] and [tex]\( B(-2,4) \)[/tex]:
[tex]\[ m_{AB} = \frac{4 - 1}{-2 - (-8)} = \frac{3}{6} = \frac{1}{2} \][/tex]
2. Calculate the slope of line [tex]\(\overleftrightarrow{CD}\)[/tex]:
Using the coordinates of points [tex]\( C(-3,-1) \)[/tex] and [tex]\( D(-6,5) \)[/tex]:
[tex]\[ m_{CD} = \frac{5 - (-1)}{-6 - (-3)} = \frac{6}{-3} = -2 \][/tex]
3. Compare the slopes to determine the relationship between the lines:
- Parallel Lines: Two lines are parallel if they have the same slope. Here, [tex]\( m_{AB} = \frac{1}{2} \)[/tex] and [tex]\( m_{CD} = -2 \)[/tex], so [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are not parallel because their slopes are different.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Check:
[tex]\[ m_{AB} \cdot m_{CD} = \left(\frac{1}{2}\right) \cdot (-2) = -1 \][/tex]
Therefore, [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are perpendicular.
Based on this analysis, only one statement is true:
B. [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are perpendicular lines.
So the correct answer is B.
1. Calculate the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
The slope formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates of points [tex]\( A(-8,1) \)[/tex] and [tex]\( B(-2,4) \)[/tex]:
[tex]\[ m_{AB} = \frac{4 - 1}{-2 - (-8)} = \frac{3}{6} = \frac{1}{2} \][/tex]
2. Calculate the slope of line [tex]\(\overleftrightarrow{CD}\)[/tex]:
Using the coordinates of points [tex]\( C(-3,-1) \)[/tex] and [tex]\( D(-6,5) \)[/tex]:
[tex]\[ m_{CD} = \frac{5 - (-1)}{-6 - (-3)} = \frac{6}{-3} = -2 \][/tex]
3. Compare the slopes to determine the relationship between the lines:
- Parallel Lines: Two lines are parallel if they have the same slope. Here, [tex]\( m_{AB} = \frac{1}{2} \)[/tex] and [tex]\( m_{CD} = -2 \)[/tex], so [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are not parallel because their slopes are different.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Check:
[tex]\[ m_{AB} \cdot m_{CD} = \left(\frac{1}{2}\right) \cdot (-2) = -1 \][/tex]
Therefore, [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are perpendicular.
Based on this analysis, only one statement is true:
B. [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{CD}\)[/tex] are perpendicular lines.
So the correct answer is B.
