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Math Help - Rotating triangle on lattice points

  1. #1
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    Rotating triangle on lattice points

    Say I have a traingle with vertices (0,0), (240,0), and (240,180). With I want to rotate this triangle about (0,0) so that the coordinates of the vertices remain integers. I want to count the number of such triangles which upon rotation have integer coordinates. What would be the algorithm for this?
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  2. #2
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    Hello decor

    Welcome to Math Help Forum!
    Quote Originally Posted by decor View Post
    Say I have a traingle with vertices (0,0), (240,0), and (240,180). With I want to rotate this triangle about (0,0) so that the coordinates of the vertices remain integers. I want to count the number of such triangles which upon rotation have integer coordinates. What would be the algorithm for this?
    We have a 3-4-5 triangle here, with sides of length 180, \,240,\, 300 units.

    In its initial position, the vertex (240,180) has coordinates (300\cos\alpha, 300\sin\alpha), where \tan\alpha = \tfrac34. If the triangle is rotated through an angle \theta anticlockwise about (0,0), then this vertex will have coordinates (300\cos(\theta+\alpha), 300\sin(\theta+\alpha))=(240\cos\theta-180\sin\theta,240\sin\theta+180\cos\theta).

    The vertex initially at (240,0) will now be at (240\cos\theta, 240\sin\theta).

    In order for these to be integer coordinates, then, all the following will need to be integers:

    • 240\cos\theta


    • 240\sin\theta


    • 180\cos\theta


    • 180\sin\theta

    Pythagorean Triples would give a general method for solving this type of problem, but with the numbers 180 and 240, the only possible values are based on 3-4-5, since no other triple has a largest number which is a factor of 180 and 240 (apart, obviously, from derivatives of this triple: 6-8-10, etc).

    So I reckon that the possible values of \theta are 0^o, \arctan(\tfrac34), \arctan(\tfrac43), 90^o,... etc.

    The first of these rotations gives coordinates:

    (240,180)\rightarrow(84,288) and (240,0)\rightarrow(192,144)

    The next gives:

    (240,180)\rightarrow(0,300) and (240,0)\rightarrow(144,192)

    ... and so on.

    Grandad
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