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Old 07-21-2011   #1
Petaminx
 
Join Date: Jul 2011
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Default Rational Coordinates of an Equilateral Triangle

Can someone list rational or integral coordinates of the vertices of an equilateral triangle?
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Old 07-23-2011   #2
MAS1

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Quote:
Originally Posted by Petaminx View Post
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.
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