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: 

VIDEO SOLUTION

STEP BY STEP SOLUTION

Given that ABCD is a parallelogram.

Therefore, AB ǁ CD and BC ǁ AD

Step 1: Since, Point P divides AB in the ratio 2:3

Therefore, if AB = a, then:

AP = a

and BP = a

Step 2: Similarly, Point Q divides CD in the ratio 4:1

Therefore, since CD = AB = a, then:

DQ = a

and QC = a

Step 3: Let’s look at Δ AOP and Δ QOC,

∠ AOP = ∠ QOC  (vertically opposite angles)

∠ OAP = ∠ QCO    (interior angles)

By AA similarity rule,

Δ AOP Δ QOC

∴ =

Step 4: Let’s start substituting values of AP and QC from previous steps:

We calculated: AP = a and QC = a

=

=

OC = OA

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

Please press Heart if you liked the solution.