Question

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.
(1 point)
Responses

(–2, –5)

(–3, 6)

(3, 2)

(3, 5)

Answer 1

[AI-generated answer — Bot GPT 3.5]

The slope of line MN is (7 - (-5)) / (4 - (-2)) = 12 / 6 = 2.

To make line MN perpendicular to line OP, the slope of line OP must be the negative reciprocal of 2, which is -1/2.
The slope of line OP is (P_y - 5) / (P_x - (-3)) = (P_y - 5) / (P_x + 3).
Thus, we need (P_y - 5) / (P_x + 3) = -1/2.
Solving this equation, we get P_y - 5 = -1/2 * (P_x + 3),
which simplifies to P_y - 5 = (-1/2)P_x - 3/2.
Rearranging, we have (-1/2)P_x - P_y = -1/2.
We can now plug in the given answer choices and see which one satisfies this equation:
A) (-1/2)(-2) - (-5) = -1/2 + 5 = 4/2 = 2, which does not satisfy the equation.
B) (-1/2)(-3) - 6 = 3/2 - 6 = -9/2, which does not satisfy the equation.
C) (-1/2)(3) - 2 = -3/2 - 2 = -7/2, which does not satisfy the equation.
D) (-1/2)(3) - 5 = -3/2 - 5 = -13/2, which does not satisfy the equation.
None of the given coordinates for P will make line MN perpendicular to line OP.