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=1/2AC`                    ................(1) 

Also, it is given that` CQ=1/4 AC   `            .............(2) 

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

`CQ/CS=((1/4)AC)/((1/2)AC)` 

Or, CQ=`1/2`CS 

Hence, Q is the midpoint of CS.  

Therefore, according to midpoint theorem in ΔCSD 

PQ || DS
If PQ || DS, we can say that QR || SB
In Δ 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.