If $A (-5, 7)$,$B (-4, -5)$,$C (-1, -6)$,and $D (4, 5)$ are the vertices of a quadrilateral,find the area of the quadrilateral $ABCD$ (in square units).

The points $A(2, 9)$,$B(a, 5)$,and $C(5, 5)$ are the vertices of a triangle $ABC$ right-angled at $B$. Find the value of $a$ and the area of $\triangle ABC$.

Find the area of the triangle whose vertices are (in square units): $(-5,-1), (3,-5), (5,2)$.

If the midpoints of the sides $\overline{AB}$,$\overline{BC}$,and $\overline{CA}$ in $\Delta ABC$ are respectively $D(1, 2)$,$E(2, 1)$,and $F(3, 3)$,then find the centroid of $\Delta ABC$.

If the vertices of $\Delta ABC$ are $A(0, 0)$,$B(4, 0)$ and $C(0, 6)$,then the coordinates of the point $P$ which is equidistant from $A, B$ and $C$ and in the plane of $\Delta ABC$ is $\ldots \ldots \ldots$

Find the circumcentre and the area of the triangle with vertices $(7,9), (10,8)$ and $(12,10)$.