1. Find the trisectional points of line joining `A(-3,-5)` and `B(-6,-8)`Solution:Trisection means dividing a line segment in three equal parts.
Let `P` and `Q` be the points of trisection of AB, then `P` divides `AB` in the ratio `1:2` and `Q` divides `AB` in the ratio `2:1`
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.
The given points are `A(-3,-5),B(-6,-8)`
`:. x_1=-3,y_1=-5,x_2=-6,y_2=-8`
and `m:n=1:2`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(1*-6+2*-3)/(1+2)`
`=(-6-6)/(3)`
`=(-12)/(3)`
`=-4`
`y=(my_2+ny_1)/(m+n)`
`=(1*-8+2*-5)/(1+2)`
`=(-8-10)/(3)`
`=(-18)/(3)`
`=-6`
Hence, the co-ordinates of the point `P` are `(-4,-6)`
Let `Q(x,y)` divides the line segment joining the points AB in the ratio `2:1`.
The given points are `A(-3,-5),B(-6,-8)`
`:. x_1=-3,y_1=-5,x_2=-6,y_2=-8`
and `m:n=2:1`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(2*-6+1*-3)/(2+1)`
`=(-12-3)/(3)`
`=(-15)/(3)`
`=-5`
`y=(my_2+ny_1)/(m+n)`
`=(2*-8+1*-5)/(2+1)`
`=(-16-5)/(3)`
`=(-21)/(3)`
`=-7`
Hence, the co-ordinates of the point `Q` are `(-5,-7)`
`:.` TriSection Points are `P(-4,-6)` and `Q(-5,-7)`
2. Find the trisectional points of line joining `A(2,-2)` and `B(-7,4)`Solution:Trisection means dividing a line segment in three equal parts.
Let `P` and `Q` be the points of trisection of AB, then `P` divides `AB` in the ratio `1:2` and `Q` divides `AB` in the ratio `2:1`
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.
The given points are `A(2,-2),B(-7,4)`
`:. x_1=2,y_1=-2,x_2=-7,y_2=4`
and `m:n=1:2`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(1*-7+2*2)/(1+2)`
`=(-7+4)/(3)`
`=(-3)/(3)`
`=-1`
`y=(my_2+ny_1)/(m+n)`
`=(1*4+2*-2)/(1+2)`
`=(4-4)/(3)`
`=(0)/(3)`
Hence, the co-ordinates of the point `P` are `(-1,0)`
Let `Q(x,y)` divides the line segment joining the points AB in the ratio `2:1`.
The given points are `A(2,-2),B(-7,4)`
`:. x_1=2,y_1=-2,x_2=-7,y_2=4`
and `m:n=2:1`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(2*-7+1*2)/(2+1)`
`=(-14+2)/(3)`
`=(-12)/(3)`
`=-4`
`y=(my_2+ny_1)/(m+n)`
`=(2*4+1*-2)/(2+1)`
`=(8-2)/(3)`
`=(6)/(3)`
`=2`
Hence, the co-ordinates of the point `Q` are `(-4,2)`
`:.` TriSection Points are `P(-1,0)` and `Q(-4,2)`
3. Find the trisectional points of line joining `A(-5,8)` and `B(10,-4)`Solution:Trisection means dividing a line segment in three equal parts.
Let `P` and `Q` be the points of trisection of AB, then `P` divides `AB` in the ratio `1:2` and `Q` divides `AB` in the ratio `2:1`
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.
The given points are `A(-5,8),B(10,-4)`
`:. x_1=-5,y_1=8,x_2=10,y_2=-4`
and `m:n=1:2`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(1*10+2*-5)/(1+2)`
`=(10-10)/(3)`
`=(0)/(3)`
`y=(my_2+ny_1)/(m+n)`
`=(1*-4+2*8)/(1+2)`
`=(-4+16)/(3)`
`=(12)/(3)`
`=4`
Hence, the co-ordinates of the point `P` are `(0,4)`
Let `Q(x,y)` divides the line segment joining the points AB in the ratio `2:1`.
The given points are `A(-5,8),B(10,-4)`
`:. x_1=-5,y_1=8,x_2=10,y_2=-4`
and `m:n=2:1`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(2*10+1*-5)/(2+1)`
`=(20-5)/(3)`
`=(15)/(3)`
`=5`
`y=(my_2+ny_1)/(m+n)`
`=(2*-4+1*8)/(2+1)`
`=(-8+8)/(3)`
`=(0)/(3)`
Hence, the co-ordinates of the point `Q` are `(5,0)`
`:.` TriSection Points are `P(0,4)` and `Q(5,0)`
4. Find the trisectional points of line joining `A(1,-2)` and `B(-3,4)`Solution:Trisection means dividing a line segment in three equal parts.
Let `P` and `Q` be the points of trisection of AB, then `P` divides `AB` in the ratio `1:2` and `Q` divides `AB` in the ratio `2:1`
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.
The given points are `A(1,-2),B(-3,4)`
`:. x_1=1,y_1=-2,x_2=-3,y_2=4`
and `m:n=1:2`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(1*-3+2*1)/(1+2)`
`=(-3+2)/(3)`
`=(-1)/(3)`
`y=(my_2+ny_1)/(m+n)`
`=(1*4+2*-2)/(1+2)`
`=(4-4)/(3)`
`=(0)/(3)`
Hence, the co-ordinates of the point `P` are `(-1/3,0)`
Let `Q(x,y)` divides the line segment joining the points AB in the ratio `2:1`.
The given points are `A(1,-2),B(-3,4)`
`:. x_1=1,y_1=-2,x_2=-3,y_2=4`
and `m:n=2:1`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(2*-3+1*1)/(2+1)`
`=(-6+1)/(3)`
`=(-5)/(3)`
`y=(my_2+ny_1)/(m+n)`
`=(2*4+1*-2)/(2+1)`
`=(8-2)/(3)`
`=(6)/(3)`
`=2`
Hence, the co-ordinates of the point `Q` are `(-5/3,2)`
`:.` TriSection Points are `P(-1/3,0)` and `Q(-5/3,2)`
5. Find the trisectional points of line joining `A(2,6)` and `B(-4,8)`Solution:Trisection means dividing a line segment in three equal parts.
Let `P` and `Q` be the points of trisection of AB, then `P` divides `AB` in the ratio `1:2` and `Q` divides `AB` in the ratio `2:1`
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.
The given points are `A(2,6),B(-4,8)`
`:. x_1=2,y_1=6,x_2=-4,y_2=8`
and `m:n=1:2`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(1*-4+2*2)/(1+2)`
`=(-4+4)/(3)`
`=(0)/(3)`
`y=(my_2+ny_1)/(m+n)`
`=(1*8+2*6)/(1+2)`
`=(8+12)/(3)`
`=(20)/(3)`
Hence, the co-ordinates of the point `P` are `(0,20/3)`
Let `Q(x,y)` divides the line segment joining the points AB in the ratio `2:1`.
The given points are `A(2,6),B(-4,8)`
`:. x_1=2,y_1=6,x_2=-4,y_2=8`
and `m:n=2:1`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(2*-4+1*2)/(2+1)`
`=(-8+2)/(3)`
`=(-6)/(3)`
`=-2`
`y=(my_2+ny_1)/(m+n)`
`=(2*8+1*6)/(2+1)`
`=(16+6)/(3)`
`=(22)/(3)`
Hence, the co-ordinates of the point `Q` are `(-2,22/3)`
`:.` TriSection Points are `P(0,20/3)` and `Q(-2,22/3)`
2. Find the trisectional points of line joining `A(3,-2)` and `B(-3,-4)`Solution:Trisection means dividing a line segment in three equal parts.
Let `P` and `Q` be the points of trisection of AB, then `P` divides `AB` in the ratio `1:2` and `Q` divides `AB` in the ratio `2:1`
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.
The given points are `A(3,-2),B(-3,-4)`
`:. x_1=3,y_1=-2,x_2=-3,y_2=-4`
and `m:n=1:2`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(1*-3+2*3)/(1+2)`
`=(-3+6)/(3)`
`=(3)/(3)`
`=1`
`y=(my_2+ny_1)/(m+n)`
`=(1*-4+2*-2)/(1+2)`
`=(-4-4)/(3)`
`=(-8)/(3)`
Hence, the co-ordinates of the point `P` are `(1,-8/3)`
Let `Q(x,y)` divides the line segment joining the points AB in the ratio `2:1`.
The given points are `A(3,-2),B(-3,-4)`
`:. x_1=3,y_1=-2,x_2=-3,y_2=-4`
and `m:n=2:1`
By section formula
`x=(mx_2+nx_1)/(m+n)`
`=(2*-3+1*3)/(2+1)`
`=(-6+3)/(3)`
`=(-3)/(3)`
`=-1`
`y=(my_2+ny_1)/(m+n)`
`=(2*-4+1*-2)/(2+1)`
`=(-8-2)/(3)`
`=(-10)/(3)`
Hence, the co-ordinates of the point `Q` are `(-1,-10/3)`
`:.` TriSection Points are `P(1,-8/3)` and `Q(-1,-10/3)`
