Find the co-ordinates of the points of trisection of the line segment AB with`A(2, 7) and B(-4, -8)`

Explanation

`A(2,7)` and `B(-4,-8)`
Let`P(x_(1),y_(1))` and `Q(x_(2),y_(2))` be the points of trisection.
`:.AP=PQ=QB` ….1
`(AP)/(PB)=(AP)/(PQ+QB)`………`(P-Q-B)`
`:.(AP)/(PB)=(AP)/(AP+AP)`…..[From 1]
`:.(AP)/(PB)=(AP)/(2AP):.(AP)/(PB)=1/2`
`:.AP:PB=1:2`
`:.P` divides segment AB in the ratio `1:2`.
By section formula,
`x_(1)=(1(-4)+2(2))/(1+2),y_(1)=(1(-8)+2(7))/(1+2)`
`:.x_(1)=(-4+4)/3,:.y_(1)=(-8+14)/3`
`:.x_(1)=0/3,:.y_(1)=6/3`
`:.x_(1)=0,:.y_(1)=2`
`:.` the coordinates of point P are (0,2).
`(AQ)/(QB)=(AP+PQ)/(QB)=.........(A-P-Q)`
`:.(AQ)/(QB)=(QB+QB)/(QB)`.......[From 1]
`:.(AQ)/(QB)=(2QB)/(QB)`
`:.(AQ)/(QB)=2/1`
`:.Q` divides segment AB in the ratio `2:1`.
By section formula,
`x_(2)=(2(-4)+1(2))/(2+1),y_(2)=(2(-8)+1(7))/(2+1)`
`:.x_(2)=(-8+2)/3,:.y_(2)=(-16+7)/3`
`:.x_(2)=(-6)/3,:.y_(2)=(-9)/3`
`:.x_(2)=-2,:.y_(2)=-3`
`:.` the coordinates of point Q are `(-2,-3)`.
The coordinates of the points of trisection of the line segment AB are (0,2) and `(-2,-3)`.