# 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:**

Let's represent a diagram according to the given question.

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 congruence criteria, triangles 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 [From equations (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.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 10

**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

**☛ Related Questions:**

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