In a kite-shaped quadrilateral $ABCD$,$AB = AD = 12 \, cm$,$CB = CD = 17 \, cm$,and $AC = 25 \, cm$. Find its area and the length of diagonal $BD$.

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 $PD$.

The perimeter of a triangle is $50 \, cm$. One side of a triangle is $4 \, cm$ longer than the smaller side and the third side is $6 \, cm$ less than twice the smaller side. Find the area of the triangle.

$A$ plot,in the shape of a quadrilateral $XYZW$,has $XY = 25 \, m$,$YZ = 60 \, m$,$ZW = 33 \, m$,$WX = 34 \, m$,and $XZ = 65 \, m$. Find its area. (in $, m^2$)

The triangular side walls of a flyover have been used for advertisements. The sides of the walls are $13 \, m$,$14 \, m$,and $15 \, m$. The advertisements yield an earning of $Rs \, 2000$ per $m^2$ a year. $A$ company hired one of its walls for $6$ months. How much rent did it pay? (in $Rs$)

The sides of a triangle measure $20 \, cm$,$24 \, cm$,and $32 \, cm$. Find its semi-perimeter. (in $, cm$)