Prove that the points A(2,4 ),B(2,6) andC(2+√3,5)are the vertices of an equilateral triangle.

Prove that the points A(2,4 ),B(2,6) andC(2+√3,5)are the vertices of

1.	Prove that the points A(2,4 ),B(2,6) andC(2+√3,5)are the vertices of an equilateral triangle.
Answer» Given : A(2, 4), B(2, 6) and C(2 +\xa0{tex}\\sqrt { 3 },{/tex}\xa05){tex}A B = \\sqrt { ( 2 - 2 ) ^ { 2 } + ( 6 - 4 ) ^ { 2 } } = \\sqrt { 0 + 2 ^ { 2 } } = \\sqrt { 4 } = 2\\ units{/tex}{tex}B C = \\sqrt { ( 2 + \\sqrt { 3 } - 2 ) ^ { 2 } + ( 5 - 6 ) ^ { 2 } }{/tex}=\xa0{tex}\\sqrt { ( \\sqrt { 3 } ) ^ { 2 } + ( - 1 ) ^ { 2 } } = \\sqrt { 3 + 1 } = \\sqrt { 4 } = 2\\ units{/tex}{tex}A C = \\sqrt { ( 2 + \\sqrt { 3 } - 2 ) ^ { 2 } + ( 5 - 4 ) ^ { 2 } }{/tex}{tex}= \\sqrt { ( \\sqrt { 3 } ) ^ { 2 } + ( 1 ) ^ { 2 } } = \\sqrt { 3 + 1 } = \\sqrt { 4 } = 2\\ units{/tex}We find that AB = BC = ACHence, the given points are the vertices of an equilateral triangle.