Question

Prove that the line joining the centres of two
intersecting circles is the perpendicular bisector of the line Joining the points of intersection​

Answer 1

To Prove:-

• AB ⊥PQ

Proof:-

Let A and B be centres of two circles intersecting at points P and Q.

____________________

In △APB and △AQB,

_____________________

AP=AQ [∵ They are radii of a circle.]

BP=BQ [∵They are radii of a circle.]

AB=AB [∵Common side]

△APB≅△AQB [By S.S.S. Criterion]

⇒∠PAB=∠QAB [By C.P.C.T.C.]

________________________________

Now, In △APR and △AQR

________________________________

AP=AQ [∵ They are radii of a circle.]

∠PAR=∠QAR. [∵∠PAB=∠QAB]

AR=AR. [∵ Common side]

△APR≅△AQR . [By S.A.S. Criterion]

⇒PR=RQ [By CPCT]

And ∠ARP=∠ARQ ...[ By C.P.C.T.C.]

Also, ∠ARP+∠ARQ=180°

∵ PQ is a straight line.

⇒∠ARP+∠ARP=180°

⇒2×∠ARP=180°

⇒∠ARP= 180°/2 =90°

Thus, AB⊥PQ