# Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ

**Solution:**

Angles in the same segment are equal.

AB is the common chord to both circles.

Since the circles are congruent, their radii are equal.

In triangles ABX and ABY

AB = AB {Common}

AX = AY {equal radii}

BX = BY {equal radii}

By SSS criteria of congruency ABX and ABY are congruent.

Hence ∠X = ∠Y {CPCT} …(1)

Now, we know that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle

Thus,

∠APB = 1/2 ∠X …(2)

∠AQB = 1/2 ∠Y …(3)

Therefore,

∠APB = ∠AQB {using …(1) …(2) and …(3)}

Consider the ΔBPQ,

∠APB = ∠AQB

This implies that ΔBPQ is an isosceles triangle as base angles are equal.

Therefore, we get BP = BQ, sides opposite to equal sides in a triangle are equal.

**Video Solution:**

## Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.6 Question 9:

**Summary:**

It is given that two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. We have proved that BP = BQ