If $D\left(\frac{-1}{2}, \frac{5}{2}\right)$,$E(7, 3)$,and $F\left(\frac{7}{2}, \frac{7}{2}\right)$ are the midpoints of the sides of $\triangle ABC$,find the area of $\triangle ABC$.

The points $(3,1)$,$(5,6)$,and $(-3,2)$ are the midpoints of the sides of a triangle. Find the vertices of the triangle.

If $(a, b)$ is the mid-point of the line segment joining the points $A (10, -6)$ and $B (k, 4)$ and $a - 2b = 18$,find the value of $k$ and the distance $AB$.

State whether the following statement is true or false. Justify your answer.
The points $(4,5), (7,6)$ and $(6,3)$ are collinear.

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 $ABC$?

Determine if the points $(1,5), (2,3),$ and $(-2,-11)$ are collinear.