$\mathrm{XYZ}$ is a right angled triangle with $\mathrm{\angle }\mathrm{X}=90°,\mathrm{XY}=9\mathrm{cm},\mathrm{YZ}=12\mathrm{cm}$. $\mathrm{OX}$ is perpendicular to $\mathrm{YZ}$. Find $\mathrm{OX}$.

The base and height of triangle are in the ratio $2:3$. If the area of the triangle is $108{\mathrm{cm}}^2$, find its base and height.

$\mathrm{ABCD}$ is a parallelogram. $\mathrm{BX}$ is the perpendicular on $\mathrm{CD}$ and $\mathrm{BY}$ is the perpendicular on $\mathrm{AD}$. If the area of the parallelogram is $1500{\mathrm{cm}}^2$, find the lengths of $\mathrm{BX}$ and $\mathrm{BY}$.

In quadrilateral $\mathrm{ABCD}$, $\mathrm{BC}$ is the diagonal, $\mathrm{DO}$ is perpendicular to $\mathrm{BC}$, and $\mathrm{CX}$ is perpendicular to $\mathrm{AB}$. If $\mathrm{AB}=25\mathrm{cm},\mathrm{BC}=16\mathrm{cm},\mathrm{CX}=8\mathrm{cm},\mathrm{DO}=5\mathrm{cm}$, find the area of the quadrilateral.

$\mathrm{ABCD}$ is a rectangle in which $\mathrm{AB}=30\mathrm{cm}$ and $\mathrm{AD}=20\mathrm{cm}$. $\mathrm{P}$ is the midpoint of $\mathrm{AD}$, $\mathrm{Q}$ is the midpoint of $\mathrm{AB}$, $\mathrm{R}$ is the midpoint of $\mathrm{BC}$, and $\mathrm{S}$ is the midpoint of $\mathrm{CD}$. If the area of the region $PQRS$ is $3k {\mathrm{cm}}^2$, find $k$.

Find the area of the shaded region.

A verandah of width $4.5\mathrm{m}$ is constructed along a room outside it. If the dimensions of the room are $18.5\mathrm{mX}14\mathrm{m}$, find:
 Area of the verandah and the cost of paving tiles in the verandah at the rate of $20$ per ${\mathrm{m}}^2$.

A rectangular field is $100\mathrm{m}$ long and $80\mathrm{m}$ wide. A path $10\mathrm{m}$ wide path is to be built inside the field all along its border. Find the area of the path.

There is a garden inside a $150$ $\mathrm{m}$ long and $100$ $\mathrm{m}$ wide rectangular park. A $20$ $\mathrm{m}$ wide path is built surrounding the garden along the boundary of the park. Find the area of the garden.