Question

in the given figure if 2 chords pq and rs of a circle with centre o intersect each other at m, such that pm=ms. Then prove that MR=MQ

Answer 1

Answer:

Step-by-step explanation:

The two chords PQ and RS of a circle with centre O intersect each other at M such that PM = MS

To prove:  MR = MQ

Construction:  Join PR and SQ

Solution:  After joining the lines you will get the triangles named as:

ΔPMR and ΔSMQ

Here we will prove further by the concept of the congruent triangles.

So let's have ΔPMR and ΔSMQ

PM = SM [Given]

∠SMQ = ∠PMR [Vertically opposite angles]

As we know that the angles subtended by the chord on any point on the boundary of the circle are same.  So,

∠MSQ = ∠MPR [For the chord QR]

ΔPMR ≅ ΔSMQ [By ASA]

∴ MR = MQ [By CPCT]

Hence proved.