Let A(1, 2) and B(11, 9) be the given points. Let the points of trisection be P and Q. Then,
AP = PQ = QB = k (say)
⇒ AQ = AP + PQ = 2k and PB = PQ + QB = 2k
∴ AP : PB = k : 2k = 1 : 2 and AQ : QB = 2k : k = 2 : 1
⇒ P divides AB internally in the ratio 1 : 2 and Q divides AB internally in the ratio 2 : 1.
∴ Coordinates of P are \(\bigg[\frac{1\times11+2\times1}{1+2},\frac{1\times9+2\times2}{1+2}\bigg]\), i.e \(\big(\frac{13}{3},\frac{13}{3}\big)\)
Coordinates of Q are \(\bigg[\frac{2\times11+1\times1}{2+1},\frac{2\times9+1\times2}{2+1}\bigg]\), i.e \(\big(\frac{23}{3},\frac{20}{3}\big)\)
Hence, the two points of trisection are \(\big(\frac{13}{3},\frac{13}{3}\big)\) and \(\big(\frac{23}{3},\frac{20}{3}\big)\).
