The area (in sq. units) bounded by $x=4$,$y=-4$ and $y=x$ is

If the midpoints of the sides of a triangle are $(0, 1), (1, 1),$ and $(1, 0)$,what is the $x$-coordinate of the incenter of the triangle?

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

If the circumcenter of the triangle formed by the points $A(a, 3)$,$B(b, 5)$,and $C(a, b)$ is $(1, 1)$,then out of all the possible coordinates of $C$,the sum of the absolute values of the distinct coordinates of $C$ is

If $A(2, -3)$ and $C(-6, 7)$ are opposite vertices of a rhombus $ABCD$, then the equation of diagonal $BD$ is

Let the area of the triangle with vertices $A(1, \alpha)$,$B(\alpha, 0)$,and $C(0, \alpha)$ be $4 \text{ sq. units}$. If the points $(\alpha, -\alpha)$,$(-\alpha, \alpha)$,and $(\alpha^2, \beta)$ are collinear,then $\beta$ is equal to: