# If 2 points (-4,3) and (0,0) are vertices of an equilateral triangle. Find the 3rd vertex.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the distance formula to find the length of the side of triangle such that:

`l = sqrt((-4-0)^2 + (3 - 0)^2) => l = sqrt(16+9) => l = 5`

You should remember that the lengths of all sides of equilateral triangle are equal, hence, using the following notation (x,y) for the missing coordinates of vertex, you may write the distance formula such that:

`5 = sqrt((x + 4)^2 + (y - 3)^2) => 25 = x^2 + 8x + 16 + y^2 - 6y + 9`

`x^2 + 8x + y^2 - 6y = 0`

`5 = sqrt((x - 0)^2 + (y - 0)^2) => 25 = x^2 + y^2`

Substituting 25 for `x^2 + y^2`  yields:

`8x - 6y = -25 => 8x = 6y - 25 => x = (6y - 25)/8`

`25 = (6y - 25)^2/64 + y^2 => 64*25 = 36y^2 - 300y + 625 + 64y^2`

`100y^2 - 300y + 625 - 1600 = 0`

`100y^2 - 300y - 975 = 0`

`20y^2 - 60y - 195 = 0`

`4y^2 - 12y - 39 = 0`

`y_(1,2) = (12 +- sqrt(144 + 624))/8`

`y_(1,2) = (12 +- sqrt768)/8 => y_(1,2) = (12 +- 16sqrt3)/8`

`y_(1,2) = (3 +- 4sqrt3)/2`

`x_(1,2) = (3(3 +- 4sqrt3) - 25)/8 => x_(1,2) = (-16 +- 4sqrt3)/8`

`x_(1,2) = (-4+-sqrt3)/2`

Hence, evaluating the  missing coordinates of the third vertex of equilateral triangle yields `((-4+-sqrt3)/2 , (3 +- 4sqrt3)/2).`