Question

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

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

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

$Letpointp\left(x,y\right)andq\left(x,y\right)point\left(p\right)dividethelinejoiningpointis1:2ratio\Rightarrow x=\frac{1\times -3+2\times 1}{1+2}=\frac{-3+2}{3}=\frac{-1}{3}\Rightarrow y=\frac{1\times 4+2\times 2}{2+1}=\frac{8}{3}andpointq\left(x,y\right)dividethelinejoiningin2:1ratio\Rightarrow y_1=\frac{2\times 4+1\times 2}{1+2}=\frac{10}{3}\Rightarrow x_1=\frac{2\times -3=1\times 1}{2+1}=\frac{-6+1}{3}=\frac{-5}{3}\therefore PointsP\left(\frac{-1}{3},\frac{8}{3}\right)andq\left(\frac{-5}{3},\frac{10}{3}\right)$