Which pairs of points would create a line that is perpendicular to the line created by the points (2,4) and (6,10)? Select all that apply.
A. (3,5) and (8,6)
B. (-1,0) and (4,3)
C. (-2,-4) and (2,2)
D. (0,2) and (-6,-4)
To determine if a line is perpendicular to another line, we need to check if the slopes of the two lines are negative reciprocals of each other.
First, let's find the slope of the line created by the points (2,4) and (6,10):
slope = (change in y-coordinates) / (change in x-coordinates)
= (10 - 4) / (6 - 2)
= 6 / 4
= 3/2
Now, let's calculate the slopes of each pair of points given in the answer choices:
A. (3,5) and (8,6):
slope = (6 - 5) / (8 - 3)
= 1 / 5
This slope is not the negative reciprocal of the slope of the given line (3/2), so this pair of points does not create a line that is perpendicular to the given line.
B. (-1,0) and (4,3):
slope = (3 - 0) / (4 - (-1))
= 3 / 5
This slope is not the negative reciprocal of the slope of the given line (3/2), so this pair of points does not create a line that is perpendicular to the given line.
C. (-2,-4) and (2,2):
slope = (2 - (-4)) / (2 - (-2))
= 6 / 4
= 3 / 2
This slope is equal to the slope of the given line (3/2), not the negative reciprocal. Therefore, this pair of points does not create a line that is perpendicular to the given line.
D. (0,2) and (-6,-4):
slope = (-4 - 2) / (-6 - 0)
= -6 / -6
= 1
This slope is the negative reciprocal of the slope of the given line (3/2), so this pair of points does create a line that is perpendicular to the given line.
Therefore, the pair of points that would create a line perpendicular to the line created by the points (2,4) and (6,10) is D. (0,2) and (-6,-4).
To determine which pairs of points would create a line perpendicular to the line passing through (2,4) and (6,10), we need to find the slope of the original line and then find the slopes of the potential perpendicular lines.
The slope of a line passing through two points (x1, y1) and (x2, y2) is calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
For the original line passing through (2,4) and (6,10):
Original line slope = (10 - 4) / (6 - 2) = 6 / 4 = 3/2
Now, let's calculate the slopes for each potential perpendicular line:
A. (3,5) and (8,6)
Perpendicular line slope = (6 - 5) / (8 - 3) = 1 / 5 ≠ 3/2
B. (-1,0) and (4,3)
Perpendicular line slope = (3 - 0) / (4 - (-1)) = 3 / 5 ≠ 3/2
C. (-2,-4) and (2,2)
Perpendicular line slope = (2 - (-4)) / (2 - (-2)) = 6 / 4 = 3/2 (This pair does create a perpendicular line)
D. (0,2) and (-6,-4)
Perpendicular line slope = (-4 - 2) / (-6 - 0) = -6 / -6 = 1 ≠ 3/2
So, the pair (C) (-2,-4) and (2,2) is the only pair of points that would create a line perpendicular to the line passing through (2,4) and (6,10).