In the given figure, $ABCD$ is a parallelogram whose diagonals intersect each other at $O$. Through $O,PQ$ is drawn then prove that $OP=OQ$.

A circle is inscribed in quadrilateral ABCD, touching sides AB, BC, CD and DA at P, Q,R and S, respectively. If $\mathrm{As}=6\mathrm{cm},$ $\mathrm{BC}=12\mathrm{cm}$ and $\mathrm{CR}=5\mathrm{cm}$, then the length of AB (in cm) is

How many quadrilaterals are there in the given figure?

Let $ABCD$ be a square. An arc of a circle with $A$ as centre and $AB$ as radius is drawn inside the square joining the points $B$ and $D$. Points $P$ on $AB, S$ on $AD, Q$ and $R$ on arc $BD$ are taken such that $PQRS$ is a square. Further suppose that $PQ$ and $RS$ are parallel to $AC$. Then, $\frac{\text{ area }PQRS}{\text{ area }ABCD}$ is

In figure, $\mathrm{ABCD}$ is a parallelogram. Point$\mathrm{E}$ is on the ray $\mathrm{AB}$ such that $\mathrm{BE}=\mathrm{AB}$, then prove that line $\mathrm{ED}$ bisects segment $\mathrm{BC}$ at point $\mathrm{F}$.

Perimeter of a parallelogram is $150 \mathrm{cm}$. One of its sides is greater than the other side by $25 \mathrm{cm}$. Find the lengths of all sides.