Question

Find the points which trisect the line segment joining the points $(0,0)$ and $(9,12)$.

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

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

The given points are $A=(0,0)$ and $B=(9,12)$

Let the point near to A be $P(P_x,P_y)$ and the point near to B be $Q(Q_x,Q_y)$

$\Rightarrow$P will divide the line AB in the ratio $1:2$ and B in the ratio $2:1$

By applying section formula

$\Rightarrow P_x=\frac{1\times 9+2\times 0}{2+1}=\frac{9}{3}=3$

$\Rightarrow P_y=\frac{1\times 12+2\times 0}{2+1}=\frac{12}{3}=4$

$\Rightarrow Q_x=\frac{2\times 9+1\times 0}{2+1}=\frac{18}{3}=6$

$\Rightarrow Q_y=\frac{2\times 12+1\times 0}{2+1}=\frac{24}{3}=8$

Hence the points that trisect line AB are $P(3,4),Q(6,8)$