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

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 $A(h, k)$,$B(1, 1)$,and $C(2, 1)$ be the vertices of a right-angled triangle with $AC$ as the hypotenuse. If the area of the triangle is $1$,then which of the following can be the set of values for $k$?

The point $(-4, 5)$ is a vertex of a square and one of its diagonals lies along the line $7x - y + 8 = 0$. The equation of the other diagonal is:

In a $\triangle ABC$,medians $AD$ and $BE$ are drawn. If $AD = 4$,$\angle DAB = \frac{\pi}{6}$,and $\angle ABE = \frac{\pi}{3}$,then the area of the $\triangle ABC$ is

$A$ triangle $ABC$ lying in the first quadrant has two vertices as $A(1, 2)$ and $B(3, 1)$. If $\angle BAC = 90^{\circ}$ and $\text{ar}(\Delta ABC) = 5\sqrt{5}$ sq. units,then the abscissa of the vertex $C$ is