Question

Q. AB is a line segment. P and Q are points on opposite sides of AB such that each of them in equidistant from the points A and B. Show that the line PQ is the perpendicular bisector of AB.

Answer 1

To prove that line PQ is the perpendicular bisector of line segment AB, we need to show that PQ passes through the midpoint of AB and that the slopes of PQ and AB are negative reciprocals of each other.

Step-by-step explanation:

Let's prove that the line PQ is the perpendicular bisector of AB:

First, observe that since P and Q are equidistant from A and B, the line PQ must pass through the midpoint of segment AB.

To show that PQ is perpendicular to AB, we need to show that the slopes of the two lines are negative reciprocals of each other.

Let's assume that the coordinates of A and B are (x1, y1) and (x2, y2), respectively. The coordinates of the midpoint M of AB are given by ((x1 + x2)/2, (y1 + y2)/2).

The slope of AB is (y2 - y1)/(x2 - x1), and the slope of PQ is -1/m, where m is the slope of AB.

Multiplying the slope of AB by -1 gives us the negative reciprocal of the slope of AB, which is the slope of PQ.

Since the line PQ passes through the midpoint of AB and has a slope that is the negative reciprocal of the slope of AB, PQ is the perpendicular bisector of AB.