Find the area of the triangle $ABC$ with vertex $A (1, -4)$ and the mid-points of the sides passing through $A$ being $(2, -1)$ and $(0, -1)$ (in $sq. units$).

Given points $A(3, 4)$ and $B(5, -2)$. Find the point $P(x, y)$ on the plane such that $PA = PB$ and the area of $\Delta PAB = 10$.

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

The coordinates of the centroid of the triangle with vertices $(2, 3)$,$(4, 7)$,and $(-3, 2)$ are:

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 $ABC$?