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?

$A$ rope by which a calf is tied is increased from $12 \ m$ to $23 \ m$. How much additional grassy ground shall it graze? (in $m^2$)

Find the area of a rhombus one of whose diagonals measures $10 \text{ cm}$ and the other $12 \text{ cm}$. (in $\text{cm}^2$)

On decreasing each side of an equilateral triangle by $2\, cm$,there is a decrease of $4\sqrt{3}\, cm^2$ in its area. The length of each side of the triangle is.........$cm$.

The ratio of the area of a sector of a circle to the area of the circle is $1:4$. If the area of the circle is $154 \, cm^2$,the perimeter of the sector is ........ $cm$.

In the given figure,$PQR$ is a triangle and quadrilateral $ABCD$ is inscribed in it. Given $QD = 2 \text{ cm}$,$QC = 5 \text{ cm}$,$CR = 3 \text{ cm}$,$BR = 4 \text{ cm}$,and $PB = 6 \text{ cm}$. What is the area (in $\text{cm}^2$) of the quadrilateral $ABCD$?