Area of a Parallelogram

$ABCD$ and $ABPQ$ are parallelogram such that the points $C, D, P, Q$ are collinear and $R$ is any point on the side BP. Show that area of $△ARQ=\frac{1}{2}$area of parallelogram $ABCD$.

$ABCD$ and $ABPQ$ are parallelogram such that the points $C, D, P, Q$ are collinear and $R$ is any point on the side $BP$. Show that area of parallelogram $ABCD=$area of parallelogram $ABPQ$.

$ABCD$ is a parallelogram and $P$ is any point on the side $CD$. Prove that area of $△APD+$area of $△BCP=$area of $△ABP$.

If $O$ is an interior point of a parallelogram $ABCD$, prove that area of $△AOB+$ area of $△OCD=$area of $△OAB+$area of $△OCD$

If $O$ is an interior point of a parallelogram $ABCD$, prove that area of $△AOB+$ area of $△OCD=\frac{1}{2}$area of parallelogram $ABCD$.

Prove that three parallelograms formed by joining the midpoints of the three sides of a triangle are equal in area.

Prove that the line segment joining the midpoints of a pair of opposite sides of a parallelogram divides it into two parallelograms of equal area.

Prove that, of all the parallelogram of given sides, parallelogram which is a rectangle has the greatest area.

If $P, Q, R$ and $S$ are respectively the midpoints of the sides of a parallelogram $ABCD$, show that area of quadrilateral$PQRS=\frac{1}{2}$(area of parallelogram $ABCD)$.

$P$ and $Q$ are any two points lying on the sides $DC$ and $AD$ respectively of a parallelogram $ABCD$. Show that area of $△APB=$area of $△BQC$.

In a parallelogram $ABCD, BC$ is of length $10 cm$. The altitudes corresponding to the sides $BC$ and $AB$ are respectively $7 cm$ and $8 cm$. Find length of $AB$.