Question

ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ = 1/4 AC. If PQ produced meets BC at R, prove that R is the midpoint of BC.

Answer 1

We know that the diagonals of a parallelogram bisect each other. 

Therefore, 

CS = 1/2 AC …(i) 

Also, it is given that CQ = 1/4 AC …(ii) 

Dividing equation (ii) by (i), we get: 

CQ/CS = 1/4AC/1/2AC 

Or, CQ = 1/2CS 

Hence, Q is the midpoint of CS.

Therefore, according to midpoint theorem in ∆CSD 

PQ || DS 

If PQ || DS, we can say that QR || SB 

In ∆ CSB, Q is midpoint of CS and QR ‖ SB.

 Applying converse of midpoint theorem , we conclude that R is the midpoint of CB. 

This completes the proof.

Comments

THANK YOU..
NICE EXPLANAATION.....

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correct