Question

Find the co-ordinates of the points of trisection of the line segment joining the points (−4, −3) and (−1, 3).

Answer 1

For a point that divides the segment joining (x₁, y₁) and (x₂, y₂) in the ratio m : n,

`x = (mx_2 + nx_1) / (m + n), y = (my_2 + ny_1) / (m + n)`

⇒ Let P be the trisection point nearer to A,

Then AP : PB = 1 : 2,

x-coordinate of P = `(1(−1) + 2(−4)) / (1 + 2)`

= `(−1 − 8) / 3`

= `−9 / 3`

= −3

y-coordinate of P = `(1 xx 3 + 2(−3)) / 3`

= `(3 − 6) / 3`

= −3 / 3 =

−1

So P = (−3, −1)

⇒ Let Q be the other trisection point, which is closer to B,

Then AQ : QB = 2 : 1,

x-coordinate of Q = `(2(−1) + 1(−4)) / 3`

= `(−2 − 4) / 3`

= `−6 / 3`

= −2

y-coordinate of Q = `(2 xx 3 + 1(−3)) / 3`

= `(6 − 3) / 3`

= `3 / 3`

= 1

So Q = (−2, 1)

Hence, the points of trisection are P(−3, −1) and Q(−2, 1).