Which of the following coordinates for P will make line MN perpendicular to line OP in the diagram below?%0D%0A%0D%0A%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A(–2, –5)%0D%0A(–2, –5)%0D%0A%0D%0A(–3, 6)%0D%0A(–3, 6)%0D%0A%0D%0A(3, 2)%0D%0A(3, 2)%0D%0A%0D%0A(3, 5)
To determine which coordinates for P will make line MN perpendicular to line OP, we need to consider the slope of the lines.
Assuming MN is represented by the line segment MN and OP is represented by the line segment OP, we can calculate the slope of each line using the formula:
slope = (y2 - y1) / (x2 - x1)
For line MN, we can use the coordinates M(-2, -5) and N(3, 2):
slope(MN) = (2 - (-5)) / (3 - (-2))
= 7 / 5
For line OP, we can use the coordinates O(0, 0) and P(x, y) where (x, y) are the coordinates of P:
slope(OP) = (y - 0) / (x - 0)
= y / x
Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of line MN should be the negative reciprocal of the slope of line OP:
7 / 5 = -1 / slope(OP)
To find the coordinates (x, y) for P that satisfy this condition, we need to solve for x and y.
Multiplying both sides of the equation by 5:
7 = (-1) / slope(OP)
Multiplying both sides by slope(OP):
7 * slope(OP) = -1
Rearranging the equation:
slope(OP) = -1 / 7
This means that the slope of line OP should be -1 / 7.
Now, let's consider the given coordinates for P:
(a) P1: (-2, -5)
(b) P2: (-3, 6)
(c) P3: (3, 2)
(d) P4: (3, 5)
We can calculate the slope for each pair of points:
(a) slope(P1) = (-5 - 0) / (-2 - 0)
= -5 / -2
= 5/2
(b) slope(P2) = (6 - 0) / (-3 - 0)
= 6 / -3
= -2
(c) slope(P3) = (2 - 0) / (3 - 0)
= 2 / 3
(d) slope(P4) = (5 - 0) / (3 - 0)
= 5 / 3
None of these slopes match the slope we obtained earlier, -1 / 7. Therefore, none of the given coordinates for P will make line MN perpendicular to line OP.
To determine which coordinates for point P will make line MN perpendicular to line OP, we need to know the slope of each line. The slope of a line can be determined using the formula:
slope = (change in y-coordinates)/(change in x-coordinates)
From the given diagram, we can see that line MN passes through the points M(2, 1) and N(4, 3). Therefore, the slope of line MN can be calculated as follows:
slope(MN) = (change in y-coordinates)/(change in x-coordinates)
= (3-1)/(4-2)
= 2/2
= 1
Now, let's consider each option for the coordinates of point P:
Option 1: (-2, -5)
The slope of line OP can be determined using the given point O(0, 0):
slope(OP) = (change in y-coordinates)/(change in x-coordinates)
= (-5-0)/(-2-0)
= -5/-2
= 5/2 ≠ -1 (not perpendicular)
Option 2: (-3, 6)
The slope of line OP with point O(0, 0):
slope(OP) = (change in y-coordinates)/(change in x-coordinates)
= (6-0)/(-3-0)
= 6/-3
= -2 ≠ -1 (not perpendicular)
Option 3: (3, 2)
The slope of line OP with point O(0, 0):
slope(OP) = (change in y-coordinates)/(change in x-coordinates)
= (2-0)/(3-0)
= 2/3 ≠ -1 (not perpendicular)
Option 4: (3, 5)
The slope of line OP with point O(0, 0):
slope(OP) = (change in y-coordinates)/(change in x-coordinates)
= (5-0)/(3-0)
= 5/3 ≠ -1 (not perpendicular)
After analyzing all the options, we can see that none of the coordinates for point P will make line MN perpendicular to line OP.
To determine which coordinates for P will make line MN perpendicular to line OP, we need to consider the slopes of the two lines.
The slope of line MN is equal to the reciprocal of the slope of line OP for the lines to be perpendicular.
Looking at the diagram, we can determine the coordinates of points M and N.
From the diagram, point M is located at (3, 2) and point N is located at (3, 5).
The slope of line MN can be determined by finding the difference in y-coordinates over the difference in x-coordinates, or (5 - 2) / (3 - 3) which equals 3/0.
However, this is undefined since we are dividing by zero. Therefore, the slope of line MN is undefined.
For line OP to be perpendicular to line MN, the slope of line OP must be the negative reciprocal of the undefined slope of line MN.
Since the undefined slope of line MN is undefined, its negative reciprocal does not exist.
Therefore, none of the given coordinates for P will make line MN perpendicular to line OP.