Question

Find the coordinates of points which trisect the line segment joining (1, –2) and (–3, 4).
$\left[4Marks\right]$

Answer 1

Let A(1, -2) and B(-3, 4) be the given points.

Let the points of trisection be P and Q. Then,

$AP=PQ=QB=\lambda$ (say).
$\left[1Mark\right]$

$\therefore PB=PQ+QB=2\lambda$ and $AQ=AP+PQ=2\lambda$

$\Rightarrow AP:PB=\lambda :2\lambda =1:2$ and

$AQ:QB=2\lambda :\lambda =2:1$

So, P divides AB internally in the ratio 1:2 while Q divides internally in the ratio 2:1.
$\left[1Mark\right]$

Thus, the coordinates of P are

$P(\frac{1\times (-3)+2\times 1}{1+2},\frac{1\times 4+2\times (-2)}{1+2})=P(-\frac{1}{3},0)$
$\left[1Mark\right]$

The coordinates of Q are

$Q(\frac{2\times (-3)+1\times 1}{2+1},\frac{2\times 4+1\times (-2)}{2+1})=Q(-\frac{5}{3},2)$
$\left[1Mark\right]$

Hence, the two points of trisection are $(-\frac{1}{3},0)$ and $(-\frac{5}{3},2)$ respectively