Which of the following coordinates for P will make line MN perpendicular to line OP in the diagram below?
A line is drawn on a coordinate grid connecting point upper M located at left-parenthesis negative 2 comma negative 5 right-parenthesis to point upper N located at left-parenthesis 4 comma 7 right-parenthesis. Point upper O is shown located at left-parenthesis negative 3 comma 5 right-parenthesis.
Responses
(–2, –5)
(–3, 6)
(3, 2)
(3, 5)
To find the coordinates for P that will make line MN perpendicular to line OP, we need to find the slope of line OP and the negative reciprocal of that slope will be the slope of line MN.
The slope of line OP can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Where (x1, y1) = (-3, 5) and (x2, y2) = (p, q) (coordinates of point P).
So the slope of line OP is:
mOP = (q - 5) / (p - (-3))
The negative reciprocal of mOP will be the slope of line MN. Let's call it mMN.
So the negative reciprocal of mOP is:
mMN = -1 / mOP = -1 / [(q - 5) / (p + 3)] = -(p + 3) / (q - 5)
To make line MN perpendicular to line OP, the slope of line MN should be the negative reciprocal of the slope of line OP. Therefore:
-(p + 3) / (q - 5) = -(7 - 5) / (4 - (-2))
Simplifying this equation will give us the value for P:
-(p + 3) / (q - 5) = -2 / 6
Cross multiplying:
-(p + 3) * 6 = -2 * (q - 5)
Simplifying further:
-6p - 18 = -2q + 10
Rearranging the equation:
-6p + 2q = 28
Now we can try each option and substitute the coordinates for P into the equation to see which one satisfies it.
Let's start with option (–2, –5):
-6(-2) + 2(-5) = 28
12 - 10 = 28
2 ≠ 28
Therefore, option (–2, –5) does not satisfy the equation.
Let's try option (–3, 6):
-6(-3) + 2(6) = 28
18 + 12 = 28
30 ≠ 28
Therefore, option (–3, 6) does not satisfy the equation.
Let's try option (3, 2):
-6(3) + 2(2) = 28
-18 + 4 = 28
-14 ≠ 28
Therefore, option (3, 2) does not satisfy the equation.
Finally, let's try option (3, 5):
-6(3) + 2(5) = 28
-18 + 10 = 28
-8 ≠ 28
Therefore, option (3, 5) does not satisfy the equation.
None of the given options satisfy the equation, so there is no coordinate for P that will make line MN perpendicular to line OP in the given diagram.