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4. Find Midpoint or Trisection points or equidistant points on X-Y axis example ( Enter your problem )
( Enter your problem )
  1. Find the coordinates of the midpoint of the line segment joining the points A(-5, 4) and B(7, -8)
  2. Find the trisectional points of line joining A(-3,-5) and B(-6,-8)
  3. Find the point on the x-axis which is equidistant from A(5,4) and B(-2,3)
  4. Find the point on the y-axis which is equidistant from A(6,5) and B(-4,3)
 		Other related methods
  1. Distance, Slope of two points
  2. Points are Collinear or Triangle or Quadrilateral form
  3. Find Ratio of line joining AB and is divided by P
  4. Find Midpoint or Trisection points or equidistant points on X-Y axis
  5. Find Centroid, Circumcenter, Area of a triangle
  6. Find the equation of a line using slope, point, X-intercept, Y-intercept
  7. Find Slope, X-intercept, Y-intercept of a line
  8. Find the equation of a line passing through point of intersection of two lines and slope or a point
  9. Find the equation of a line passing through a point and parallel or perpendicular to Line-2 or point-2 and point-3
  10. Find the equation of a line passing through point of intersection of Line-1, Line-2 and parallel or perpendicular to Line-3
  11. For two lines, find Angle, intersection point and determine if parallel or perpendicular lines
  12. Reflection of points about x-axis, y-axis, origin
1. Find the coordinates of the midpoint of the line segment joining the points A(-5, 4) and B(7, -8)
(Previous example)		3. Find the point on the x-axis which is equidistant from A(5,4) and B(-2,3)
(Next example)

2. Find the trisectional points of line joining A(-3,-5) and B(-6,-8)


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)`





This material is intended as a summary. Use your textbook for detail explanation.
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1. Find the coordinates of the midpoint of the line segment joining the points A(-5, 4) and B(7, -8)
(Previous example)		3. Find the point on the x-axis which is equidistant from A(5,4) and B(-2,3)
(Next example)


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