Two vertices of an equilateral triangle are (a,-a) and (-a,a). Find the third.

Question

I need the solution. Answers are (a(sqrt3),a(sqrt3)) and (-a(sqrt3),-a(sqrt3))

Answer 1

Let the equilateral triangle be PQR such that p(-a, a); Q(a,-a); and R(x,y)
The point R is equidistant from P and Q and therefore must be on the perpendicular bisector of the line segment PQ.   Note that the origin (0,0) is also equidistant from P and Q and therefore the line segment OR { ( 0,0) (x,y)} represents a part or the line that is the perpendicular bisector of PQ.

The slope of PQ is =  a - -a =    2a   = -1.
                                -a -a       -2a
If s₁ and s₂ are the slopes of two perpendicular lines, then their products is - 1
s1 x s2 = -1
So -1 x the slpope of OR is -1; therefore the slope of OR is 1 and since it passes through the origin the equation of the line OR is y = x.
The triangle is equilateral and therefore all the sides are congruent or of equal length.

The length d, of a line is given by d = the square root of the sum of (x ₂ - x₁)² and (y ₂ - y₁)²
This application does not allow me to draw diagrams or to insert the symbols that I would like and so the effort to explain the answer is more difficult than it needs to be. I hope you can follow.

Squaring both sides of the equation: ( d )² = (the square root of the sum of (x₂ -x₁)² and (y ₂ - y₁)²
So d² = (x₂ -x₁)² + (y₂ - y₁)²
This means that the squares of the lengths of the sides of the triangle are equal.

PQ² = QR² = PR²
PQ² = (-a - a)² + (a - - a)²
= ( -2a)² + ( 2a)²
= 4a² + 4a² = 8a²
Note carefully that the point R has coordinates (x,y); but it lies on the line y = x.

To simplify our calculation, we will make the coordinates of R to be (x,x)
QR² = (a - x)² + ( -a - x)² = PQ² = 8a²
= (a - x)² + {- (a + x) }²
= a² - 2ax + x² + a² + 2ax + x²
= 2x² + 2a² = 2( x² + a² )
PQ² = QR²
8a² = 2( x² + a² )
4a² = x² + a²
3a² = x²
Taking the square roots of both sides
x = a root 3
The coordinates or R are ( a root 3, a root 3)
Note that another triangle can be drawn such that its vertex is the reflection of R about the origin. Its coordinates are      (- a root 3, -a root 3)