What coordinate of the third vertex creates an equilateral triangle?

Question

Suppose two vertices of a triangle are (0,x) and (x,0)

Answer 1

Make a drawing using a>0 as the x and y intercepts.

Draw the ray y=x through the first quadrant.

The third vertex must lie on that ray.

The base of the equilateral joins the two intercepts; its length is a√2.

Imagine a circular arc centered at (0,a) and having radius a√2, intersecting the ray.

That point of intersection is the third vertex.

To determine it, solve the system

  y = x

  x² + (y-a)² = [a√2]²

 

Before proceeding, make sure that you understand why this system's solution leads to both coordinates of the third vertex.

 

My answer is x = y = ½(1 + √3)a.

 

I have omitted any consideration of the third vertex appearing in the third quadrant.