In parallelogram ABCD. P is mid-point of AB. CP and BD intersect each other at point O. If area of $Δ P O B = 40 c m^{2}$ , find a if $O C : O P = a : 2$ .

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Solution

Join PD in the above figure.
It is given that in a parallelogram ABCD. P is the midpoint of AB. CP and BD intersects at each other at point O and we know that the corresponding sides of a parallelogram are proportional , therefore,

$Δ P O B \sim Δ C O D$

Since P is the midpoint of AB thus,

$\frac{P B}{C P} = \frac{O P}{O C}$

$\Rightarrow$ $\frac{O P}{O C} = \frac{1}{2}$

$\Rightarrow$ $\frac{O P}{O C} = 1 : 2$

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