Given two points $A(3, 4)$ and $B(5, -2)$. If $PA = PB$ and the area of $\Delta PAB = 10$ square units,find the coordinates of point $P$.

If the equation of the base of an equilateral triangle is $x+y=6$ and the opposite vertex is the point $(-1, -1)$,then the area of the triangle is equal to $k$ sq. units. Find the value of $k$.

In a rectangle $ABCD$,points $X$ and $Y$ are the mid-points of $AD$ and $DC$,respectively. Lines $BX$ and $CD$ when extended intersect at $E$,lines $BY$ and $AD$ when extended intersect at $F$. If the area of $ABCD$ is $60$,then the area of $\triangle BEF$ is

In a $\triangle ABC$,$D$ and $E$ are points on $AB$ and $AC$ respectively such that $DE \parallel BC$. Suppose $BE$ and $CD$ intersect at $O$. If the areas of $\triangle ADE$ and $\triangle ODE$ are $3$ and $1$ respectively,find the area of $\triangle ABC$.

The area of the triangle formed by the lines $x + y - 3 = 0$,$x - 3y + 9 = 0$,and $3x - 2y + 1 = 0$ is:

Suppose $ABCD$ is a trapezium whose sides and height are integers and $AB$ is parallel to $CD$. If the area of $ABCD$ is $12$ and the sides are distinct,then $|AB-CD|$ is: