The sides of a quadrilateral $ABCD$ are $6 \, cm$,$8 \, cm$,$12 \, cm$,and $14 \, cm$ (taken in order) respectively,and the angle between the first two sides is a right angle. Find its area in $cm^2$.

State whether the following statement is True or False and justify your answer:
The area of $\Delta ABC$ is $8 \, cm^2$ in which $AB = AC = 4 \, cm$ and $\angle A = 90^{\circ}$.

Find the area of the trapezium $PQRS$ given in the figure.

In a quadrilateral $ABCD$,$AB = 9 \, cm$,$BC = 40 \, cm$,$CD = 28 \, cm$,$DA = 15 \, cm$,and $\angle B = 90^{\circ}$. Find its area. (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 $PD$.

$A$ field is in the shape of a trapezium having parallel sides $90 \, m$ and $30 \, m$. These sides meet the third side at right angles. The length of the fourth side is $100 \, m$. If it costs $\operatorname{Rs} 4$ to plough $1 \, m^2$ of the field,find the total cost of ploughing the field (in $\operatorname{Rs}$).