**Q) **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.

**Ans: **

Given that ABCD is a parallelogram. Therefore, AB ǁ CD and BC ǁ AD

Since, Point P divides AB in the ratio 2:3

Therefore, if AB = a, then AP = a and BP = a

Since, Point Q divides CD in the ratio 4:1

Therefore, since CD = AB = a, then DQ = a and QC = a

Let’s look at Δ AOP and Δ QOC,

∠ AOP = ∠ QOC (vertically opposite angles)

∠ OAP = ∠ QCO (interior angles)

Therefore, Δ AOP Δ QOC

Hence, =

, =

=

OC = OA

**Therefore, it is proved that OC is half of OA.**