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# Show that the points a (1,1),B(-1,-1) and C(-√3,√3) are the vertex of an equilateral triangle each of whose sides is 2 routes to unit

Vikas TU
14149 Points
one year ago
Dear student
Any triangle is said to be equilateral triangle when all sides of triangle be equal.
so, use distance formula and find length of all sides of given triangle.
if (x1, y1) and (x2, y2) two points are given then, distance between them = √{(x1 - x2)² +(y1 - y2)²}
Let A = (-1, -1) , B = (1, 1) and C = (-√3, √3)
length of side AB = √{(-1 - 1)² + (-1 - 1)²}
= √{(-2)² + (-2)²}
= √{4 + 4} = √8
=2√2
length of side BC = √{(-1 + √3)² + (-1 - √3)²}
= √{√3² + 1² - 2√3 + √3² + 1² + 2√3}
= √{3 + 1 + 3 + 1}
= √{8}
= 2√2
length of side CA = √{(-√3 + 1)² + (√3 + 1)²}
= √{√3² + 1² - 2√3 + √3² + 1² + 2√3}
= √{3 + 1 + 3 + 1 }
= √8
= 2√2
here we see that length of side AB = length of side BC = length of side CA
so, ABC is an equilateral triangle.