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 $k.$

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 $\Delta ABC$.

State whether the following statement is true or false. Justify your answer.
Point $P (0, -7)$ is the point of intersection of the $y$-axis and the perpendicular bisector of the line segment joining the points $A (-1, 0)$ and $B (7, -6)$.

In $\Delta ABC$,if $A(b, c)$,$B(-a, 0)$,and $C(a, 0)$,and $D$ is the midpoint of $\overline{BC}$,then prove that $AB^2 + AC^2 = 2(AD^2 + BD^2)$.

If $(-8, -4), (-2, 4)$ and $(5, a)$ are collinear points,then find the value of $a$.

Show that,for every $a \in R$,the area of the triangle having the vertices $A(5, a)$,$B(2, 5)$,and $C(2, 3)$ is $3$ square units.