Method and examples
Coordinate Geometry
Method
0. All methods
1. Distance, Slope of two points
2. Points (3 or 4) are Collinear or Triangle or Quadrilateral form
3. Find Ratio of line joining AB and is divided by P
4. 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
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. Angle between the two lines
12. Reflection of points about x-axis, y-axis, origin
1.1 Distance, Slope of two points
1. Find distance between the points `A(7,8)` and `B(1,0)`

2. Find slope of the line joining points `A(7,8)` and `B(1,0)`

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`A(7,8),B(1,0)` `A(2,5),B(8,5)` `A(1,2),B(-3,5)` `A(-2,0),B(4,8)` `A(-2,0),B(-8,8)` `A(5,1),B(1,-2)` `A(6,4),B(-1,5)` `A(2,3),B(7,6)` `A(-3,4),B(5,-6)` `A(0,7),B(5,-2)` `A(0,0),B(-4,-6)` `A(3,5),B(6,4)`
1.2 Find the value of x or y
3. If distance between A(1,x) and B(-3,5) is 5, find the value of x.

Distance = ,
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`distance=5`, `A(7,8),B(x,0)` `distance=5`, `A(1,x),B(-3,5)` `distance=10`, `A(x,0),B(4,8)` `distance=10`, `A(-2,x),B(-8,8)` `distance=5`, `A(5,1),B(x,-2)` `distance=6`, `A(6,x),B(-1,5)` `distance=7`, `A(2,x),B(7,6)`
4. If slope of A(-2,x) and B(5,-7) is -1, find the value of x.

Slope = ,
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`slope=1.33`,`A(7,8),B(x,0)` `slope=-1`,`A(-2,x),B(5,-7)` `slope=3/5`,`A(2,3),B(x,6)` `slope=-5/4`,`A(-3,4),B(5,x)` `slope=-9/5`,`A(0,x),B(5,-2)` `slope=3/2`,`A(x,0),B(-4,-6)`

2. Points are Collinear or Triangle or Quadrilateral form
Find `A(0,0), B(2,2), C(0,4), D(-2,2)` are vertices of a square or not

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Find
0. auto detect
1. Collinear points (ABC)
2. right angle triangle (ABC)
3. equilateral triangle (ABC)
4. isosceles triangle (ABC)
5. Collinear points (ABCD)
6. square (ABCD)
7. rectangle (ABCD)
8. rhombus (ABCD)
9. parallelogram (ABCD)
`A(-1,-1),B(1,5)``,C(2,8)` are vertices of are collinear points (ABC) `A(0,-1),B(3,5)``,C(5,9)` are vertices of are collinear points (ABC) `A(2,8),B(1,5)``,C(0,2)` are vertices of are collinear points (ABC) `A(2,5),B(8,5)``,C(5,10.196152)` are vertices of an equilateral triangle (ABC) `A(2,2),B(-2,4)``,C(2,6)` are vertices of an isosceless triangle (ABC) `A(0,0),B(0,3)``,C(4,0)` are vertices of a right angle triangle (ABC) `A(-2,-2),B(-1,2)``,C(3,1)` are vertices of a right angle triangle (ABC) `A(-3,2),B(1,2)``,C(-3,5)` are vertices of a right angle triangle (ABC) `A(0,0),B(2,0)``,C(-4,0)``,D(-2,0)` are vertices of are collinear points (ABCD) `A(2,3),B(7,4)``,C(8,7)``,D(3,6)` are vertices of a parallelogram (ABCD) `A(1,5),B(1,4)``,C(-1,3)``,D(-1,4)` are vertices of a parallelogram (ABCD) `A(1,-1),B(-2,2)``,C(4,8)``,D(7,5)` are vertices of a rectangle (ABCD) `A(0,-4),B(6,2)``,C(3,5)``,D(-3,-1)` are vertices of a rectangle (ABCD) `A(3,0),B(4,5)``,C(-1,4)``,D(-2,-1)` are vertices of a rhombus (ABCD) `A(3,2),B(5,4)``,C(3,6)``,D(1,4)` are vertices of a square (ABCD) `A(0,0),B(2,2)``,C(0,4)``,D(-2,2)` are vertices of a square (ABCD)

3. Find Ratio
1. Find ratio of line joining A(5,12) and B(2,9) is divided by a point P(3,10)

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`A(1,-3),B(3,5)``,P(6,17)` `A(2,8),B(6,14)``,P(12,23)` `A(5,12),B(2,9)``,P(3,10)` `A(5,13),B(1,4)``,P(17/5,47/5)`
2. Find a point which divides the line joining A(-4,1) and B(17,10) in the ratio 1:2

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ratio =
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Externally
`A(5,13),B(1,4)``,m:n=2:3` `A(-4,1),B(17,10)``,m:n=1:2` `A(5,12),B(2,9)``,m:n=2:1` `A(2,8),B(6,14)``,m:n=5:3` Externally `A(1,-3),B(3,5)``,m:n=5:3` Externally
3. Find the ratio in which the x-axis divides the join of `A(2,1)` and `B(7,6)`? Also find the coordinates of the point of intersection.

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`A(2,1),B(7,6)` divided by x-axis `A(2,-4),B(-3,6)` divided by x-axis `A(2,-4),B(-3,6)` divided by y-axis
4. Find the ratio in which the point `P(x,4)` divides the join of `A(2,1)` and `B(7,6)`? Also find the value of `x`.

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`P(x,4),A(2,1),B(7,6)` `P(x,0),A(2,-4),B(-3,6)` `P(0,y),A(2,-4),B(-3,6)`

4. Find Midpoint or Trisection points or equidistant points on X-Y axis
1. Find the midpoint of line joining points `A(-2,3)` and `B(5,4)`

2. Find the trisection points of line joining points `A(-2,3)` and `B(5,4)`

3. Find the point on the x-axis which is equidistant from `A(-2,3)` and `B(5,4)`

4. Find the point on the y-axis which is equidistant from `A(-2,3)` and `B(5,4)`

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`A(-4,1),B(17,10)` `A(-2,3),B(5,4)` `A(-3,4),B(5,-6)` `A(0,7),B(5,-2)` `A(0,0),B(-4,-6)` `A(3,5),B(6,4)`

5. Find Centroid, Circumcenter, Area of a triangle
1. Find the centroid of a triangle whose vertices are `A(4,1),B(-2,-3),C(6,7)`

2. Find the circumcentre of a triangle whose vertices are `A(4,1),B(-2,-3),C(6,7)`

3. Find the area of a triangle whose vertices are `A(4,1),B(-2,-3),C(6,7)`

4. Find `A(4,1),B(-2,-3),C(6,7)` are collinear points or not (using area finding method)

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`A(2,5),B(4,9)``,C(6,4)` `A(-1,-1),B(1,5)``,C(2,8)` `A(-3,2),B(1,2)``,C(-3,5)` `A(0,-1),B(3,5)``,C(5,9)` `A(4,1),B(-2,-3)``,C(6,7)`

6. Find the equation of a line using slope, point, X-intercept, Y-intercept
1. Find the equation of a line passing through point `A(5,4)` and having slope `1/2`

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Slope :
slope`=2`,`A(0,3)` slope`=1/4`,`A(0,0)` slope`=1/2`,`A(5,4)`
2. Find the equation of a line passing through two points `A(3,5)` and `B(6,4)`

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`A(7,8),B(1,0)` `A(6,4),B(-1,5)` `A(2,3),B(7,6)` `A(-3,4),B(5,-6)` `A(0,7),B(5,-2)` `A(0,0),B(-4,-6)` `A(3,5),B(6,4)`
3. Find the equation of a line having slope = 2 and Y-intercept = 3

Slope : ;
Y-intercept :
slope`=1/2`,Y-intercept`=5` slope`=1/4`,Y-intercept`=0` slope`=2`,Y-intercept`=3`
4. Find the equation of a line having X-intercept = 3 and Y-intercept = -5

X-intercept : ;
Y-intercept :
X-intercept`=2`,Y-intercept`=-2` X-intercept`=-5/3`,Y-intercept`=5` X-intercept`=-3/5`,Y-intercept`=-3/2` X-intercept`=3`,Y-intercept`=-5`

7. Find Slope, X-intercept, Y-intercept of a line
1. Find Slope and Y-intercept of a line 7y-4x+9=0

2. Find X-intercept and Y-intercept of a line 7y-4x+9=0

3. Find Slope, X-intercept and Y-intercept of a line 7y-4x+9=0

Line :
Line`:2x+3y+4=0` Line`:3x+6y-8=0` Line`:4x+5y+7=0` Line`:3x-2y-12=0` Line`:7y-4x+9=0` Line`:5x+2y-11=0` Line`:3x-y+11=0` Line`:4x-3y+2=0`
4. Find Slope, X-intercept and Y-intercept of the line joining the points `A(3,-5)` and `B(-7,9)`

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`A(7,8),B(1,0)` `A(6,4),B(-1,5)` `A(2,3),B(7,6)` `A(-3,4),B(5,-6)` `A(0,7),B(5,-2)` `A(0,0),B(-4,-6)` `A(3,5),B(6,4)` `A(3,-5),B(-7,9)`

8. Find the equation of a line passing through point of intersection of two lines and slope or a point
1. Find the equation of a line passing through point of intersection of lines `2x+3y+4=0` and `3x+6y-8=0` and having slope = 2

Line-1 : ,
Line-2 : ,
Slope :
Line-1`:x-4y+18=0`,Line-2`:x+y-12=0`,slope`=2` Line-1`:2x+3y+4=0`,Line-2`:3x+6y-8=0`,slope`=2` Line-1`:x=3y`,Line-2`:3x=2y+7`,slope`=-1/2`
2. Find the equation of a line passing through point of intersection of lines `4x+5y+7=0` and `3x-2y-12=0` and point `A(3,1)`

Line-1 : ,
Line-2 : ,
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Line-1`:x+y+1=0`,Line-2`:3x+y-5=0`,`A(1,-3)` Line-1`:4x+5y+7=0`,Line-2`:3x-2y-12=0`,`A(3,1)`

9. Find the equation of a line passing through a point and parallel or perpendicular to Line-2
1. Find the equation of a line passing through point `A(5,5)` and parallel to the line `2x+3y+4=0`

2. Find the equation of a line passing through point `A(5,5)` and perpendicular to the line `2x+3y+4=0`

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Line-2 :
`A(5,7)`,Line`:2x+3y+5=0` `A(4,2)`,Line`:3x-2y=5` `A(5,5)`,Line`:2x+3y+4=0`
3. Find the equation of a line passing through point `A(5,5)` and parallel to the line passing `B(1,-2)` and `C(-5,2)`

4. Find the equation of a line passing through point `A(5,5)` and perpendicular to the line passing `B(1,-2)` and `C(-5,2)`

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`A(5,7),B(-1,-1),C(-4,1)` `A(4,2),B(1,-1),C(3,2)` `A(5,5),B(1,-2),C(-5,2)`

12. Reflection of points about x-axis, y-axis, origin
Find Reflection of pointsA(0,0),B(2,2),C(0,4),D(-2,2) and Reflection about X,Y,O

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Reflection =
`A(-2,-2)``,B(-1,2)``,C(3,1)` and Reflection about X `A(2,3)``,B(7,4)``,C(8,7)``,D(3,6)` and Reflection about Y `A(1,-1)``,B(-2,2)``,C(4,8)``,D(7,5)` and Reflection about O `A(3,0)``,B(4,5)``,C(-1,4)``,D(-2,-1)` and Reflection about X,Y `A(3,2)``,B(5,4)``,C(3,6)``,D(1,4)` and Reflection about Y,X `A(-1,-1)``,B(1,5)``,C(2,8)` and Reflection about X `A(-3,2)``,B(1,2)``,C(-3,5)` and Reflection about Y `A(0,-1)``,B(3,5)``,C(5,9)` and Reflection about X `A(2,8)``,B(1,5)``,C(0,2)` and Reflection about Y `A(0,0)``,B(2,2)``,C(0,4)``,D(-2,2)` and Reflection about X,Y,O

Solution
Solution provided by AtoZmath.com

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