Consider the following points:
M (-6, 6) N (6, 4) O (-6, -4) P (8, -8)
Which of the following is true of lines MP and NO ?
Responses
A They are parallel.They are parallel.
B They intersect at (0, 0).They intersect at (0, 0).
C They intersect at (2, 6).They intersect at (2, 6).
D They intersect at (-3, 4).
To determine if the lines MP and NO are parallel or if they intersect at a specific point, we can calculate the slopes of the lines.
The slope of line MP can be determined using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points M (-6, 6) and P (8, -8):
m_MP = (-8 - 6) / (8 - (-6))
m_MP = -14 / 14
m_MP = -1
The slope of line NO can be determined using the points N (6, 4) and O (-6, -4):
m_NO = (-4 - 4) / (-6 - 6)
m_NO = -8 / -12
m_NO = 2/3
Since the slopes of the two lines are different (-1 and 2/3), the lines MP and NO are not parallel.
To check if the lines intersect at a specific point, we can use the equation for the line in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
For line MP, using point M (-6, 6) and slope -1:
6 = -1 * (-6) + b
6 = 6 + b
b = 0
So the equation for line MP is y = -x + 0, which simplifies to y = -x.
For line NO, using point N (6, 4) and slope 2/3:
4 = (2/3) * 6 + b
4 = 4 + b
b = 0
So the equation for line NO is y = (2/3)x + 0, which simplifies to y = (2/3)x.
From the equations, we can see that both lines have the same y-intercept of 0 and have different slopes. Therefore, the lines MP and NO intersect at the point (0, 0).
The correct answer is B: They intersect at (0, 0).