The sides of a triangle are $35 \text{ cm}$,$54 \text{ cm}$,and $61 \text{ cm}$,respectively. The length of its longest altitude is:

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 $PD$.

In parallelogram $ABCD$,$AB = 9 \, cm$,$BC = 10 \, cm$ and $AC = 17 \, cm$,then find its area. (in $, cm^2$)

$A$ rhombus-shaped sheet with a perimeter of $40 \, cm$ and one diagonal of $12 \, cm$ is painted on both sides at the rate of $Rs \, 5$ per $cm^2$. Find the cost of painting (in $Rs$).

For a plank in the shape of a parallelogram,the adjacent sides measure $12 \, cm$ and $17 \, cm$,and one of its diagonals measures $25 \, cm$. Find the cost of painting the plank at the rate of Rs. $3$ per $cm^2$.

Calculate the area of the shaded region in the figure.