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

Given that ABCD is a parallelogram, we know that opposite sides are equal and parallel.

Let's denote the points where P divides AB and Q divides DC as A and C respectively.

Since P divides AB in the ratio 2:3, we can say that AP:PB = 2:3. Similarly, since Q divides DC in the ratio 4:1, we can say that DQ:QC = 4:1.

In a parallelogram, diagonals bisect each other. Therefore, the point where the diagonals intersect (let's call it O) divides AC and BD in a 1:1 ratio.

This means that AO = OC and BO = OD.

Since AP:PB = 2:3, we can say that AO:OB = 2:3.

Similarly, since DQ:QC = 4:1, we can say that DO:OC = 4:1.

Therefore, by comparing the ratios, we can conclude that OC is half of OA.

This proves that OC is half of OA in the given parallelogram ABCD.