The line segment joining $A(-3, 5)$ and $B(2, 5)$ intersects............

The points $A(x_{1}, y_{1})$,$B(x_{2}, y_{2})$ and $C(x_{3}, y_{3})$ are the vertices of $\triangle ABC$.
$(i)$ The median from $A$ meets $BC$ at $D$. Find the coordinates of the point $D$.
$(ii)$ Find the coordinates of the point $P$ on $AD$ such that $AP : PD = 2 : 1$.
$(iii)$ Find the coordinates of points $Q$ and $R$ on medians $BE$ and $CF$,respectively,such that $BQ : QE = 2 : 1$ and $CR : RF = 2 : 1$.
$(iv)$ What are the coordinates of the centroid of the triangle $ABC$?

Show that the points $A(-2, 1)$,$B(2, -2)$,and $C(5, 2)$ are the vertices of a right-angled triangle.

If the distance between points $A(a, 2)$ and $B(3, -5)$ is $\sqrt{53},$ find the possible values of $a$.

Find the points of trisection of the line segment joining $(-2,-1)$ and $(7,8)$.

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