Question

Find the co ordinates of points which trisect the line segment joining (1, -2) and (-3, 4)

• A. $(-\frac{1}{3},0)and(-\frac{5}{3},2)$
• B. $(\frac{1}{3},0)and(-\frac{5}{3},2)$
• C. $(-\frac{1}{3},0)and(\frac{5}{3},2)$
• D. none of the above

Answer 1

The correct option is A $(-\frac{1}{3},0)and(-\frac{5}{3},2)$

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)
$\therefore$ PB = PQ + QB = 2$\lambda$ and AQ = AP + PQ = 2$\lambda$
$\Rightarrow$ AP : PB = $\lambda :2\lambda =1:2$ and AQ : QB = $\lambda :2\lambda =2:1$
So P divides AB internally in the ratio 1 : 2 while Q divides internally in the ratio 2 : 1
$\therefore$ the co-ordinates of P are $(\frac{1\times -3+2\times 1}{1+2},\frac{1\times 4+2\times -2}{1+2})or(-\frac{1}{3},0)$
and the co-ordinates of Q are $(\frac{2\times -3+1\times 1}{2+1},\frac{2\times 4+1\times (-2)}{2+1})or(-\frac{5}{3},2)$
Hence, the points of trisection are $(-\frac{1}{3},0)and(-\frac{5}{3},2)$