If the area of a triangle with vertices $(a, 5), (6, 7)$ and $(2, 3)$ is $10$ square units,then $a = \dots$ (where $a \in N$).

State whether the following statement is true or false. Justify your answer.
Points $A(3, 1)$,$B(12, -2)$,and $C(0, 2)$ cannot be the vertices of a triangle.

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 $\ldots \ldots \ldots$

If the vertices of $\Delta ABC$ are $A(9,5), B(6,7)$ and $C(3,3)$,then the length of median $\overline{BE}$ is.......

If $A(x_1, y_1), B(x_2, y_2),$ and $C(x_3, y_3)$ are the vertices of $\Delta ABC,$ then the area of $\Delta ABC$ is equal to:

Two vertices of a triangle are $A(2, 1)$ and $B(3, -2)$. The third vertex is $C(x, y)$,where $y = x + 3$. If the area of the triangle is $5$,find the coordinates of the third vertex.