In the given figure,$PB$ is one-third of $AB$ and $BQ$ is one-third of $BC.$ If the area of $BPDQ$ is $20 \, cm^{2},$ then what is the area (in $cm^{2}$) of $ABCD$?

The area of a square is $1024\, cm^2$. What is the ratio of the length to the breadth of a rectangle whose length is twice the side of the square and breadth is $12\, cm$ less than the side of this square?

$D$ and $E$ are points on sides $AB$ and $AC$ of $\Delta ABC$. $DE$ is parallel to $BC$. If $AD:DB = 2:3$,what is the ratio of the area of $\Delta ADE$ to the area of quadrilateral $BDEC$?

The sides of a triangle are in the ratio $3:4:5$. If its perimeter is $36 \text{ cm}$,then the area of the triangle is ...... $\text{cm}^2$.

$A$ hemispherical bowl of internal diameter $54 \ cm$ contains a liquid. The liquid is to be filled in cylindrical bottles of radius $3 \ cm$ and height $9 \ cm$. How many bottles are required to empty the bowl?

$ABC$ is a triangle. $AB = 5 \text{ cm}$,$AC = \sqrt{41} \text{ cm}$,and $BC = 8 \text{ cm}$. $AD$ is perpendicular to $BC$. What is the area (in $\text{cm}^2$) of triangle $ABD$?