The vertices of $\Delta ABC$ are $A(3, 4)$,$B(0, 0)$,and $C(6, 0)$. Then the length of the median $\overline{AD}$ is:

Find the value of $a$,if the distance between the points $A(-3, -14)$ and $B(a, -5)$ is $9$ units.

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 $ABC$?

Prove that the points $A(1, -2)$,$B(2, 3)$,$C(-3, 2)$,and $D(-4, -3)$ are the vertices of a rhombus.

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 three vertices of a parallelogram $ABCD$ are $A(3,2)$,$B(4,5)$,and $C(2,3)$,then find the coordinates of the fourth vertex $D$.