Find the coordinates of the points which divide the line segment joining `A(-2,2) and B(2,8)` into four equal parts.

Explanation

Let P, Q and R be the poins on line segment AB such that
`" "AP=PQ=QR=RB`
Let `" "AP=PQ=QR=RB=k`

Now, `(AP)/(PB)=(k)/(3k)=(1)/(3)`
Therefore, P divides AB internally in the ration 1 : 3.
`because" Internally ratio"=((m_(1)x_(2)+m_(2)x_(1))/(m_(1)+m_(2)), (m_(1)y_(2)+m_(2)y_(1))/(m_(1)+m_(2)))`
`" "P=((1xx2+3(-2))/(1+3), (1xx8+3xx2)/(1+3))=((2-6)/(4), (8+6)/(4))=((-4)/(4), (14)/(4))=(-1, (7)/(2))`
Again, `" "(AR)/(RB)=(3k)/(k)=(3)/(1)`
Therefore, R divides AB internally in the ratio 3 : 1.
`therefore" "R=((3xx2+1xx(-2))/(3+1), (3xx8+1xx2)/(3+1))`
`" "=((6-2)/(4), (24+2)/(4))=((4)/(4), (26)/(4))=(1, (13)/(2))`
Also, `" "(AQ)/(QB)=(2k)/(2k)=(1)/(1)`
`therefore` Q is the mid-point of AB.
`therefore" "Q=((-2+2)/(2), (2+8)/(2))[because "mid-point"=((x_(1)+x_(2))/(2), (y_(1)+y_(2))/(2))]`
`i.e., " "Q=((0)/(2), (10)/(2))=(0, 5)`
So, required points are `(-1, (7)/(2)), (0, 5) and (1, (13)/(2))`.