# Determine the coordinates of points along which the ends A (1, -3), B (4.3) divided into three equal parts.

sciencesolve | Certified Educator

You need to divide the length of segment AB into three equal parts, hence, you need to find the length of the segment AB using distance equation such that:

`AB = sqrt((x_B - x_A)^2 + (y_B - y_A)^2)`

`AB = sqrt((4 - 1)^2 + (3 - (-3))^2)`

`AB = sqrt(9 + 36) => AB = sqrt45 => AB = 3sqrt5`

You need to divide the length of segment AB by 3 such that:

`(AB)/3 = (3sqrt5)/3 => (AB)/3 = sqrt 5`

Hence, each segment has a length of `sqrt 5` .

Since the points M and N divide the segment AB in three equal parts, you need to find the coordinates of M, such that:

`AM = sqrt((x_M - x_A)^2 + (y_M - y_A)^2)`

`AM = sqrt((x_M - 1)^2 + (y_M + 3)^2)`

Substituting `sqrt 5` for AM yields:

`sqrt 5 = sqrt((x_M - 1)^2 + (y_M + 3)^2) => 5 = (x_M - 1)^2 + (y_M + 3)^2`

`MN = sqrt 5 = sqrt((x_N - x_M)^2 + (y_N - y_M)^2)`

`5 = (x_N - x_M)^2 + (y_N - y_M)^2`

`NB = sqrt 5 = sqrt((4 - x_N)^2 + (3 - y_N)^2)`

`5 = (4 - x_N)^2 + (3 - y_N)^2`

`(x_M - 1)^2 + (y_M + 3)^2 = (x_N - x_M)^2 + (y_N - y_M)^2 = (4 - x_N)^2 + (3 - y_N)^2`

`(x_M)^2 - 2x_M + 1 + (y_M)^2 + 6y_M + 9 = (x_N)^2 - 2x_N*x_M + (x_M)^2 + (y_N)^2 - 2y_N*y_M + (y_M)^2`

`6y_M - 2x_M + 10 = -2x_N*x_M - 2y_N*y_M`

`(x_N)^2 - 2x_N*x_M + (x_M)^2 + (y_N)^2 - 2y_N*y_M + (y_M)^2 = 16 - 8x_N + (x_N)^2 + 9 - 6y_N + (y_N)^2`

`- 2x_N*x_M - 2y_N*y_M = - 8x_N- 6y_N + 25`

`6y_M - 2x_M + 10 = - 8x_N- 6y_N + 25 => 6(y_M + y_N) + 8x_N - 2x_M = 25 - 10 = 15`

`(x_M)^2 - 2x_M + 1 + (y_M)^2 + 6y_M + 9 = 16 - 8x_N + (x_N)^2 + 9 - 6y_N + (y_N)^2`

`(x_M)^2 - (x_N)^2 + (y_M)^2 - (y_N)^2 - 2x_M + 8x_N + 6y_M + 6y_N = 0`

`(x_M)^2 - (x_N)^2 + (y_M)^2 - (y_N)^2 + 15 - 15 = 0`

`(x_M)^2 - (x_N)^2 = (y_N)^2 - (y_M)^2`

Hence, evaluating the relation between the coordinates of the points M and N that divide the segment AB into three equal parts yields `(x_M)^2 - (x_N)^2 = (y_N)^2 - (y_M)^2.`

vaaruni | Student

We are require to find the co-ordinates of the point dividing the line AB Joining the points A(1,-3)  and B(4,-3)

Let C and  D are the  points which divides the line AB into 3 equal parts.

Then,  AC = CD = DB  i.e the point C divides the line AB in the ratio  1:2

[ AC=1  and  CB = 2 ]

Using section formula that is the co-ordinate of a point (x,y) dividing a line AB joining the two points  A(x1,y1)  and  B(x2,y2)  in the ratio  m:n  is Given by the  formula :-

x = (m*x1+n*x2)/(m+n)   and    y = (m*y1+n*y2)/(m+n)

x = (1*4 + 2*1) / (1+2) = (4+2)/3 = 6/3 = 2 ,         x = 2 <--

y = ( 1*(-3)+2(-3)) /(1+2) = (-3-6)/3 = -9/3 = -3,   y = -3 <--

co-ordinate of the point   C(x,y) = (2,-3)

Finding co-ordinate of the point   D  :

The point D divides the line AB in the ratio  2:1  [ AD= 2 , BD= 1]

Using the same section formula

x = (m*x1+n*x2)/(m+n)   and    y = (m*y1+n*y2)/(m+n)

x= ( 2*4+1*1) / (1+2) = (8+1)/3 = 9/3 = 3 ,             x = 3 <--

y = (2*(-3) ) +1*(-3))/(1+2) = (-6-3)/3 = -9/3 = -3   y = -3 <--

Hence

The co-ordinates of the points are :

(2, -3)  and (3, -3)    <--- Answer