Question

An equilateral triangle has 2 vertex at points (3,4) and (-2,3), find the coordinates of the third vertex

Answer 1

Given two vertices of equilateral triangle are
A(3,4) & B(-2,3)

Let the third vertex of equilateral triangle be P(x,y)
Distance between points A & B
= √[(3+2)^2 + (4-3)^2
= √(25+1)
= √26

Distance between P & A = Distance between P & B = Distance between A & B

√[(x-3)^2 + (y-4)^2] = √[(x+2)^2 + (y-3)^2] = √26

squaring on both sides ;

(x-3)^2 + (y-4)^2 = (x+2)^2 + (y-3)^2 = 26
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x^2+9-6x +y^2+16-8y = 26
x^2+y^2-6x-8y = 1
x^2 + y^2 = 6x+8y+1 -------------(1)
------------------------------------
26 = x^2+y^2+4x-6y+11
x^2+y^2+4x-6y = 15
x^2 + y^2 = -4x+6y+15 ----------(2)
-----------------------------------
x^2+9-6x+y^2+16-8y = x^2+y^2+4x-6y+11
10x+3y = 12 ----------------(3)
-----------------------------------
In equation (1) & (2) RHS is equal , therefore ;
6x+8y+1 = -4x+6y+15
10x+2y = 14 --------------(4)

____________________
Solving equation (3) & (4)

y = -2

x = 18
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Hence the third vertex of the triangle is (-2,18)