Question

Find the points of trisection of the line segment joining the points:

5, −6 and (−7, 5),

Answer 1

The coordinates of a point which divided two points `(x_1,y_1)` and `(x_2, y_2)` internally in the ratio m:n is given by the formula,

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

The points of trisection of a line are the points which divide the line into the ratio 1: 2.

Here we are asked to find the points of trisection of the line segment joining the points A(5,−6) and B(−7,5).

So we need to find the points which divide the line joining these two points in the ratio 1: 2 and 2: 1.

Let P(x, y) be the point which divides the line joining ‘AB’ in the ratio 1 : 2.

`(x,y) = (((1-(-7) + 2(5))/(1 +2 ))","((1(5) + 2(-6))/(1+ 2)))`

`(x, y) = (1, 7/3)`

Let Q(e, d) be the point which divides the line joining ‘AB’ in the ratio 2 : 1.

(e, d) = `(((1(5) + 2(-7))/(1 + 2))", "((1(-6) +  2(5))/(1 + 2))`

`(e,d) = (-3, 4/3)`

Therefore the points of trisection of the line joining the given points are `(1,7/3) and (-3, 4/3)`