Question

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

Answer 1

Trisection means dividing a line segment in three equal parts or dividing a line segment in the ratio $1:2$ and $2:1$

Given points: $x_1=3,x_2=-3,y_1=-2,y_2=-4$

CASE :- 1

$m:n=1:2$

$=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

$=(\frac{1(-3)+\left(3\right)}{2+1},\frac{1(-4)+2(-2)}{2+1})$

$=(\frac{-3+6}{3},\frac{-4-4}{3})$

$=(-1,\frac{-8}{3})$

CASE :- 2

$m:n=2:1$

$=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

$=(\frac{2(-3)+1\left(3\right)}{2+1},\frac{2(-4)+1(-2)}{2+1})$

$=(\frac{-6+3}{3},\frac{-8-2}{3})$

$=(\frac{-3}{3},\frac{-10}{3})$

$=(-1,\frac{-10}{3})$