The sides of $\Delta PQR$ measure $13 \, cm$, $14 \, cm$, and $15 \, cm$. Find its area in $cm^2$.

Find the area of the parallelogram given in the figure. Also,find the length of the altitude from vertex $A$ on the side $DC.$

The shape of a field is a triangle. The measurements of the sides of the field are $51 \,m$,$52 \,m$ and $53 \,m$. Find its area. (in $,m^2$)

Write True or False and justify your answer.
The area of a triangle with base $4 \, \text{cm}$ and height $6 \, \text{cm}$ is $24 \, \text{cm}^2$.

In quadrilateral $ABCD$, one of its diagonals $AC$ measures $20 \, cm$. The altitudes on $AC$ from vertices $B$ and $D$ are $8 \, cm$ and $12 \, cm$ respectively. Find the area of quadrilateral $ABCD$ in $cm^2$.

The dimensions of a rectangle $ABCD$ are $51 \, cm \times 25 \, cm$. $A$ trapezium $PQCD$ with its parallel sides $QC$ and $PD$ in the ratio $9:8$ is cut off from the rectangle as shown in the figure. If the area of the trapezium $PQCD$ is $\frac{5}{6}$ th part of the area of the rectangle,find the lengths $QC$ and $PD$.