Question

the line segment joining the points 3,-1 and -6,5 is trisected . Find the coordintes of the points of trisection?​

Answer 1

Given

Points (3,-1) and (-6, 5)

To find

The trisection points of the line segment

Solution

Let P be one point that divides the above line segment in the ratio $(1:2)$

$(m,n)=(1,2)$

The coordinates of P are

$\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)\\ =\left(\dfrac{1\times -6+2\times 3}{1+2},\dfrac{1\times 5+2\times -1}{1+2}\right)\\ =\left(0,1\right)$

Let Q be one point that divides the above line segment in the ratio $(2:1)$

$(m,n)=(2,1)$

The coordinates of Q are

$\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)\\ =\left(\dfrac{2\times -6+1\times 3}{2+1}, \dfrac{2\times 5+1\times -1}{2+1}\right)\\ =\left(-\dfrac{9}{3},\dfrac{9}{3}\right)\\ =\left(-3,3\right)$

The points of trisection are $(0,1)$ and $(-3,3)$.