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Solution
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Solution provided by AtoZmath.com
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Determine if the points are Collinear points calculator
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 1. Determine if the points `A(1,5), B(2,3), C(-2,-11)` are collinear points
2. Determine if the points `A(1,-3), B(2,-5), C(-4,7)` are collinear points
3. Determine if the points `A(-1,-1), B(1,5), C(2,8)` are collinear points
4. Determine if the points `A(0,-1), B(3,5), C(5,9)` are collinear points
5. Determine if the points `A(2,8), B(1,5), C(0,2)` are collinear points
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Example1. Determine if the points `A(1,5), B(2,3), C(-2,-11)` are collinear pointsSolution: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 pointsSolution: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 pointsSolution: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 pointsSolution: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 pointsSolution: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 
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