If the vertices of $\Delta ABC$ are $A(x_{1}, y_{1})$,$B(x_{2}, y_{2})$,and $C(x_{3}, y_{3})$,then the centroid of $\Delta ABC$ is........

Given $P-R-Q$ and if $\frac{PR}{PQ} = \frac{2}{7}$,then $R$ divides $\overline{PQ}$ from $Q$ in the internal division ratio $\ldots \ldots \ldots \ldots$

Using the distance formula,show that $(4, 3)$,$(5, 1)$,and $(1, 9)$ are collinear.

If the mid-point of the line segment joining the points $A (3, 4)$ and $B (k, 6)$ is $P (x, y)$ and $x+y-10=0,$ find the value of $k.$

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 $ABC$?

The distance between $A(\cos \theta, 0)$ and $B(0, \sin \theta)$ is ...........