Points P, Q, and R in that order are dividing line segment joining A (1,6) and B(5, -2) in four equal parts. Find the coordinates of P, Q and R.

#### Solution

The given points are A(1,6) and B(5,-2) .

Then, P (x, y). is a point that devices the line AB in the ratio 1:3

By the section formula:

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

`⇒ x = (1 xx 5+3xx1)/(1+3) = y = (1 xx(-2) +3xx6)/(1+3)`

` ⇒ x = (5+3)/4 , y = (-2+18)/4`

` ⇒ x = 8/4 , y = 16/4 `

x= 2 and y = 4

Therefore, the coordinates of point P are (2,4)

Let Q be the mid-point of AB

Then, , Q( x y)

` x= (x_1+x_2)/2 , y = (y_1+y_2)/2`

`⇒x = (1+5)/2 , y= (6+(-2))/2`

` ⇒ x =6/2 , y = 4/2 `

⇒ x = 3, y= 2

Therefore, the coordinates of Q are( 3,2)

Let , R( x y) be a point that divides AB in the ratio 3:1

Then, by the section formula:

` x = ((3x_2 +nx_1))/((m+n)) , y = ((my_2 +ny_1))/((m+n))`

` ⇒ x = ((3xx5+1xx1))/(3+1) , y = ((3 xx (-2) +1 xx6))/(3+1)`

`⇒ x = (15+1)/4 , y = (-6+6)/4`

`⇒ x = 16/4 , y= 0/4`

⇒ x = 4 and y = 0

Therefore, the coordinates of R are (4,0) .

Hence, the coordinates of point P, Q and R are (2,4) , (3,2) and (4,0) respectively