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Question
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
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