Question

ABCD is a parallelogram. Point P divides AB in the ratio 2:3 and point Q divides DC in the ratio 4:1. Prove that OC is half of OA.

Answer 1

ABCD is a parallelogram.

AB = DC = a

Point P divides AB in the ratio 2:3

AP = `2/5a`, BP = `3/5a`

Point Q divides DC in the ratio 4:1.

DQ = `4/5a`, CQ = `1/5a`

ΔAPO ∼ ΔCQO  ...[AA similarity]

`(AP)/(CQ) = (PO)/(QO) = (AO)/(CO)`

`(AO)/(CO) = (2/5a)/(1/5a) = 2/1`

`\implies` OC = `1/2 OA`