Rational Coordinates of an Equilateral Triangle

[Petaminx]

Can someone list rational or integral coordinates of the vertices of an equilateral triangle?

[MAS1]

Originally Posted by Petaminx

Can someone list rational or integral coordinates of the vertices of an equilateral triangle?

Let's say you have an equilateral triangle with sides length s, where s is a rational or integral number, and vertices A, B, and C.

Let's put one of the vertices, A, at the origin of the x-y plane. So we have A (0,0).

Let's locate vertex C on the positive x-axis. It's location is C (s,0).

To find the location of vertex B, since the triangle is equilateral, then the x coordinate of B must be halfway between the x coordinates of A and C, or s/2. The y coordinate can be found by noting that a perpendicular bisector from B forms a 30-60-90 triangle with B at the 30 deg angle and A at the 60 deg angle at the origin. The ratios of side lengths for this type of triangle is 1:sqrt(3):2. Therefore since the short side has length s/2, the the y coordinate is sqrt(3)*s/2. So the coordinates for B are (s/2,sqrt(3)*s/2).

A: (0,0)
B: (s/2,sqrt(3)*s/2)
C: (s,0)

Hope this helps.