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

Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side $a$ such that $a^{2}+11 a+3(m_{1}^{2}+m_{2}^{2})=220$. If one vertex of the square is $(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$,where $\alpha \in(0, \frac{\pi}{2})$ and the equation of one diagonal is $(\cos \alpha-\sin \alpha) x +(\sin \alpha+\cos \alpha) y =10$,then $72(\sin ^{4} \alpha+\cos ^{4} \alpha)+a^{2}-3 a+13$ is equal to.

The area of the triangle enclosed by the straight lines $x = 0$,$y = 0$ and $x + 2y + 3 = 0$ in sq. units is:

The triangle with vertices $A(2, 4)$,$B(2, 6)$,and $C(2 + \sqrt{3}, 5)$ is a . . . .

Two sides of a square with an area of $25$ square units are given by the equations $3x - 4y = 0$ and $4x + 3y = 0$. The equations of the other two sides are:

The sides of a triangle are $3x + 4y$,$4x + 3y$,and $5x + 5y$ units,where $x, y > 0$. The triangle is: