Question

A parallelogram ABCD has P the mid-point of Dc and Q a point of Ac such that

CQ = `[1]/[4]`AC. PQ produced meets BC at R.

Prove that
(i)R is the midpoint of BC
(ii) PR = `[1]/[2]` DB

Answer 1

For help, we draw the diagonal BD as shown below

The diagonal AC and BD cuts at point X.

We know that the diagonal of a parallelogram intersect equally with each other. Therefore

AX = CX and BX = DX

Given,
CQ = `[1]/[4]`AC

CQ = `[1]/[4]` x 2CX

CQ = `[1]/[2]`CX

Therefore Q is the midpoint of CX.

(i) For triangle CDX PQ || DX or PR || BD
Since for triangle CBX
Q is the midpoint of CX and QR || BX. Therefore R is the midpoint of BC

(ii) For triangle BCD
As P and R are the mid-point of CD and BC, therefore  PR = `[1]/[2]` DB