Question

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

Answer 1

It is given that

P and Q are equidistant from A and B that is

, and

We are asked to show that line PO is perpendicular bisector of line AB.

First of all we will show that ΔAQP and ΔQBP are congruent to each other and ultimately we get the result.

Consider the triangles AQP and QBP in which

AP=BP, AQ=BQ, PQ=PQ

So by SSS property we have

Implies that

Now consider the triangles ΔAPC and ΔPCB in which

And

So by SAS criterion we find that,

So this implies that AC=BC and

But

Hence PQ is perpendicular bisector of AB.