Points of Trisection of a Line Segment

A line segment $AB$ is increased along its length by $25\%$ by producing it to $C$ on the side of $B$. If $A$ and $B$ have the coordinates $(-2,-3)$ and $(2,1)$ respectively, then find the coordinates of $C$.

Find the coordinates of point $P$ on the line segment joining $A(1,2)$ and $B(6,7)$ in such a way that $AP=\frac{2}{5}AB$.

In what ratio does the point $P(2,-5)$ divide the line segment joining $A(-3,5)$ and $B(4,-9)$?
 

Find the coordinates of the point which divides the line segment joining the points $A(4,-3) \mathrm{and} B(9,7)$ in the ratio $3:2$.

Find the coordinates of points of trisection of the line segment joining the point $(6,-9)$ and the origin.

Find the coordinates of the points of trisection of the line segment joining $(4,-1)$ and $(-2,-3)$.

If $AP:PB=1:2$ and $AQ:QB=2:1$, then find the values of $AP:AB$ and $AQ:AB$.

What happens when the values in the section formula, $m=n=1?$ Can you identify it with a result already proved?

Using section formula, show that the points $A(7,-5), B(9,-3)$ and $C(13,1)$ are collinear.

The line segment joining $A(6,3)$ and $B(-1,-4)$ is doubled in length by adding half of $AB$ to each end. Find the coordinates of the new end points.

Find the coordinates of the points of trisection of the line segment joining the points $A(-5,6)$ and $B(4,-3)$.