Two equal circles intersect at P and Q

We know that two circles will congruent if they have equal radii

From the figure, we know that if two  chords are equal then the corresponding arcs are congruent

We know that PQ is the common chord in both the circles

So their corresponding arcs are equal

It can be written as

Arc PCQ=arc PDQ

We know that the congruent arcs have the same degree

So we get

∠QAP=∠QBP

We know that the base angles of an isosceles triangle are equal

So we get

QA=AB

Therefore, it is proved that QA=QB

13. Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB.

Let C (O, r) and C(O', r) be two equal circles. clearly, C(O, r) ≅ C(O', r).

Since PQ is a common chord of two congruent circles.Therefore,arc PCQ = arc PDQ

⇒ ∠ QAP = ∠ QBP

Thus, in ΔQAB, we have∠ QAP = ∠ QBP⇒ QA = QB

Hence proved.

Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB.

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