The quadrilateral formed by the vertices $A(-1, 1)$,$B(0, -3)$,$C(5, 2)$,and $D(4, 6)$ 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:

If $A(4,7)$,$B(-7,8)$,and $C(1,2)$ are the vertices of $\triangle ABC$,then the equation of the perpendicular bisector of the side $AB$ is

If a pair of perpendicular lines through the origin together with the straight line $2x + 3y = 6$ form an isosceles triangle,then the area of that triangle (in sq units) is

Let one of the sides of a triangle be $17 \text{ cm}$ and the sum of all the sides of the triangle be $40 \text{ cm}$. If the sum of two adjacent sides is $35 \text{ cm}$, then the area (in $\text{cm}^2$) of the triangle is (in $\sqrt{2}$)

Let $A(1, 1)$,$B(-4, 3)$,and $C(-2, -5)$ be the vertices of a triangle $ABC$. Let $P$ be a point on the side $BC$,and let $\Delta_{1}$ and $\Delta_{2}$ be the areas of triangle $APB$ and triangle $ABC$,respectively. If $\Delta_{1} : \Delta_{2} = 4 : 7$,then find the area enclosed by the lines $AP$,$AC$,and the $x$-axis.