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

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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.`

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**