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
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