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Solution
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Solution provided by AtoZmath.com
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Show that the points are the vertices of an equilateral triangle calculator
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 1. Show that the points `A(1,1), B(-1,-1), C(-1.732051,1.732051)` are vertices of an equilateral triangle
2. Show that the points `A(2,5), B(8,5), C(5,10.196152)` are vertices of an equilateral triangle
3. Show that the points `A(1,2), B(1,6), C(4.4641,4)` are vertices of an equilateral triangle
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Example1. Show that the points `A(1,1), B(-1,-1), C(-1.732051,1.732051)` are vertices of an equilateral triangleSolution: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)` A Triangle, in which all sides are equal, is called an equilateral triangle The given points are `A(1,1),B(-1,-1),C(-1.7321,1.7321)` `AB=sqrt((-1-1)^2+(-1-1)^2)` `=sqrt((-2)^2+(-2)^2)` `=sqrt(4+4)` `=sqrt(8)` `:. AB=2sqrt(2)` `BC=sqrt((-1.7321+1)^2+(1.7321+1)^2)` `=sqrt((-0.7321)^2+(2.7321)^2)` `=sqrt(0.5359+7.4641)` `=sqrt(8)` `:. BC=2sqrt(2)` `AC=sqrt((-1.7321-1)^2+(1.7321-1)^2)` `=sqrt((-2.7321)^2+(0.7321)^2)` `=sqrt(7.4641+0.5359)` `=sqrt(8)` `:. AC=2sqrt(2)` Here `AB=BC=AC` `:.` ABC is an equilateral triangle 
2. Show that the points `A(2,5), B(8,5), C(5,10.196152)` are vertices of an equilateral triangleSolution: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)` A Triangle, in which all sides are equal, is called an equilateral triangle The given points are `A(2,5),B(8,5),C(5,10.1962)` `AB=sqrt((8-2)^2+(5-5)^2)` `=sqrt((6)^2+(0)^2)` `=sqrt(36+0)` `=sqrt(36)` `:. AB=6` `BC=sqrt((5-8)^2+(10.1962-5)^2)` `=sqrt((-3)^2+(5.1962)^2)` `=sqrt(9+27)` `=sqrt(36)` `:. BC=6` `AC=sqrt((5-2)^2+(10.1962-5)^2)` `=sqrt((3)^2+(5.1962)^2)` `=sqrt(9+27)` `=sqrt(36)` `:. AC=6` Here `AB=BC=AC` `:.` ABC is an equilateral triangle 
3. Show that the points `A(1,2), B(1,6), C(4.4641,4)` are vertices of an equilateral triangleSolution: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)` A Triangle, in which all sides are equal, is called an equilateral triangle The given points are `A(1,2),B(1,6),C(4.4641,4)` `AB=sqrt((1-1)^2+(6-2)^2)` `=sqrt((0)^2+(4)^2)` `=sqrt(0+16)` `=sqrt(16)` `:. AB=4` `BC=sqrt((4.4641-1)^2+(4-6)^2)` `=sqrt((3.4641)^2+(-2)^2)` `=sqrt(12+4)` `=sqrt(16)` `:. BC=4` `AC=sqrt((4.4641-1)^2+(4-2)^2)` `=sqrt((3.4641)^2+(2)^2)` `=sqrt(12+4)` `=sqrt(16)` `:. AC=4` Here `AB=BC=AC` `:.` ABC is an equilateral triangle  |
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