Final answer:
The midpoint M of PQ is defined by averaging the coordinates of P and Q. The slopes of CM and PQ are then found and shown to be negative reciprocals, proving CM is perpendicular to PQ.
Step-by-step explanation:
To show that CM is perpendicular to PQ, we first need to find the coordinates of the midpoint M of the line segment PQ. The midpoint M of a segment with endpoints P(-2, 3) and Q(4, -1) is found by averaging the x and y coordinates of P and Q, which gives us M(1, 1). Next, we determine the slopes of CM and PQ to establish the perpendicularity.
The slope of PQ is calculated by (y2 - y1) / (x2 - x1), which gives us (-1 - 3) / (4 - (-2)) = -4 / 6 = -2 / 3. The slope of CM is found using the coordinates of M(1, 1) and C(-1, -2), which gives us (-2 - 1) / (-1 - 1) = -3 / -2 = 3 / 2.
The slopes of two perpendicular lines are negative reciprocals of each other. Since the slope of CM (3/2) is the negative reciprocal of the slope of PQ (-2/3), CM is therefore perpendicular to PQ.