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2. Points are Collinear or Triangle or Quadrilateral form example ( Enter your problem )
( Enter your problem )
  1. A(-1,-1), B(1,5), C(2,8), Find points are collinear points or triangle
  2. Determine if the points A(1,5), B(2,3), C(-2,-11) are collinear points
  3. Show that the points A(-3,0), B(1,-3), C(4,1) are vertices of a right angle triangle
  4. Show that the points A(1,1), B(-1,-1), C(-1.732051,1.732051) are vertices of an equilateral triangle
  5. Show that the points A(7,10), B(-2,5), C(3,-4) are vertices of an isosceles triangle
  6. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
  7. Show that the points A(1,2), B(5,4), C(3,8), D(-1,6) are vertices of a square
  8. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
  9. Show that the points A(3,0), B(4,5), C(-1,4), D(-2,-1) are vertices of a rhombus
  10. Show that the points A(-3,-2), B(5,-2), C(9,3), D(1,3) are vertices of a parallelogram
 		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. A(-1,-1), B(1,5), C(2,8), Find points are collinear points or triangle
(Previous example)		3. Show that the points A(-3,0), B(1,-3), C(4,1) are vertices of a right angle triangle
(Next example)

2. Determine if the points A(1,5), B(2,3), C(-2,-11) are collinear points


1. Determine if the points `A(1,5), B(2,3), C(-2,-11)` are collinear points

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


The given points are `A(1,5),B(2,3),C(-2,-11)`

`AB=sqrt((2-1)^2+(3-5)^2)`

`=sqrt((1)^2+(-2)^2)`

`=sqrt(1+4)`

`=sqrt(5)`

`:. AB=sqrt(5)`


`BC=sqrt((-2-2)^2+(-11-3)^2)`

`=sqrt((-4)^2+(-14)^2)`

`=sqrt(16+196)`

`=sqrt(212)`

`:. BC=2sqrt(53)`


`AC=sqrt((-2-1)^2+(-11-5)^2)`

`=sqrt((-3)^2+(-16)^2)`

`=sqrt(9+256)`

`=sqrt(265)`

`:. AC=sqrt(265)`


As, AC > AB and AC > BC
If points A, B and C are collinear then AB + BC = AC
But `sqrt(5)+2sqrt(53)=16.7963!=sqrt(265)`

`:.` A,B,C are not collinear points



2. Determine if the points `A(1,-3), B(2,-5), C(-4,7)` are collinear points

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


The given points are `A(1,-3),B(2,-5),C(-4,7)`

`AB=sqrt((2-1)^2+(-5+3)^2)`

`=sqrt((1)^2+(-2)^2)`

`=sqrt(1+4)`

`=sqrt(5)`

`:. AB=sqrt(5)`


`BC=sqrt((-4-2)^2+(7+5)^2)`

`=sqrt((-6)^2+(12)^2)`

`=sqrt(36+144)`

`=sqrt(180)`

`:. BC=6sqrt(5)`


`AC=sqrt((-4-1)^2+(7+3)^2)`

`=sqrt((-5)^2+(10)^2)`

`=sqrt(25+100)`

`=sqrt(125)`

`:. AC=5sqrt(5)`


As, BC > AB and BC > AC
If points A, B and C are collinear then AB + AC = BC
Here `sqrt(5)+5sqrt(5)=6sqrt(5)`

`:.` A,B,C are collinear points



3. Determine if the points `A(-1,-1), B(1,5), C(2,8)` are collinear points

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


The given points are `A(-1,-1),B(1,5),C(2,8)`

`AB=sqrt((1+1)^2+(5+1)^2)`

`=sqrt((2)^2+(6)^2)`

`=sqrt(4+36)`

`=sqrt(40)`

`:. AB=2sqrt(10)`


`BC=sqrt((2-1)^2+(8-5)^2)`

`=sqrt((1)^2+(3)^2)`

`=sqrt(1+9)`

`=sqrt(10)`

`:. BC=sqrt(10)`


`AC=sqrt((2+1)^2+(8+1)^2)`

`=sqrt((3)^2+(9)^2)`

`=sqrt(9+81)`

`=sqrt(90)`

`:. AC=3sqrt(10)`


As, AC > AB and AC > BC
If points A, B and C are collinear then AB + BC = AC
Here `2sqrt(10)+sqrt(10)=3sqrt(10)`

`:.` A,B,C are collinear points



4. Determine if the points `A(0,-1), B(3,5), C(5,9)` are collinear points

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


The given points are `A(0,-1),B(3,5),C(5,9)`

`AB=sqrt((3-0)^2+(5+1)^2)`

`=sqrt((3)^2+(6)^2)`

`=sqrt(9+36)`

`=sqrt(45)`

`:. AB=3sqrt(5)`


`BC=sqrt((5-3)^2+(9-5)^2)`

`=sqrt((2)^2+(4)^2)`

`=sqrt(4+16)`

`=sqrt(20)`

`:. BC=2sqrt(5)`


`AC=sqrt((5-0)^2+(9+1)^2)`

`=sqrt((5)^2+(10)^2)`

`=sqrt(25+100)`

`=sqrt(125)`

`:. AC=5sqrt(5)`


As, AC > AB and AC > BC
If points A, B and C are collinear then AB + BC = AC
Here `3sqrt(5)+2sqrt(5)=5sqrt(5)`

`:.` A,B,C are collinear points



5. Determine if the points `A(2,8), B(1,5), C(0,2)` are collinear points

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


The given points are `A(2,8),B(1,5),C(0,2)`

`AB=sqrt((1-2)^2+(5-8)^2)`

`=sqrt((-1)^2+(-3)^2)`

`=sqrt(1+9)`

`=sqrt(10)`

`:. AB=sqrt(10)`


`BC=sqrt((0-1)^2+(2-5)^2)`

`=sqrt((-1)^2+(-3)^2)`

`=sqrt(1+9)`

`=sqrt(10)`

`:. BC=sqrt(10)`


`AC=sqrt((0-2)^2+(2-8)^2)`

`=sqrt((-2)^2+(-6)^2)`

`=sqrt(4+36)`

`=sqrt(40)`

`:. AC=2sqrt(10)`


As, AC > AB and AC > BC
If points A, B and C are collinear then AB + BC = AC
Here `sqrt(10)+sqrt(10)=2sqrt(10)`

`:.` A,B,C are collinear points





This material is intended as a summary. Use your textbook for detail explanation.
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1. A(-1,-1), B(1,5), C(2,8), Find points are collinear points or triangle
(Previous example)		3. Show that the points A(-3,0), B(1,-3), C(4,1) are vertices of a right angle triangle
(Next example)


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