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.
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
∠APB = 1/2 ∠X …(2)
∠AQB = 1/2 ∠Y …(3)
∠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.
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
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:
- Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its center. If the distance between AB and CD is 6 cm, find the radius of the circle.
- The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the center, what is the distance of the other chord from the center ?
- Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the center.
- Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.