Question

Find the co-ordinates of the points of trisection of the line segment joining the points $P(-3,4)$ and $Q(4,5)$

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

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

If $(x_1,y_1)$ and $(x_2,y_2)$ are the points of trisection then they will divide the line segment $PQ$ in the ratio $2:1.$ and $1:2$ respectively.

By Section Formula,

$(x_1,y_1)=(\frac{1\times -3+2\times 4}{1+2},\frac{1\times 4+2\times 5}{1+2})$

$=(\frac{5}{3},\frac{14}{3})$

Similarly,

$(x_2,y_2)=(\frac{1\times 4+2\times -3}{1+2},\frac{1\times 5+2\times 4}{1+2})$

$=(-\frac{2}{3},\frac{13}{3})$

So, the points of trisection are $(\frac{5}{3},\frac{14}{3})$ and $(\frac{-2}{3},\frac{13}{3})$.