Triangulation equation?

I have three points in an X,Y coordinate system. Those points are A, B, and C forming a triangle. I want to know the coordinates of A. The coordinates of B and C, the distance AB, and the angle A between AB and AC are all known.

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  • Dave
    Lv 6
    7 years ago
    Favorite Answer

    By Law of Cosines

    (BC)^2=(AB)^2+(AC)^2 -2(AC)(AB)cos(<A)

    We know AB, the angle A, and we can easily find BC knowing the coordinates.

    solving the quadratic equation,

    0=(AC)^2 -2(AB)cos(<A)(AC)+((AB)^2-(BC)^2)

    we know the distance AC.

    (AB)^2=(Bx-Ax)^2+(By-Ay)^2

    (AC)^2=(Cx-Ax)^2+(Cy-Ay)^2

    using these two equations, we can easily solve for Ax and Ay.

  • 7 years ago

    Let B (X₂, Y₂), C (X₃, Y₃); let the given angle A be θ degree and distance AB = d. Point A(h, k) or we need find h and k. For this- We need to solve following two simultaneous equations .

    ([{Y₂-k)/(X₂-h)} - {Y₃-k)/(X₃-h)}]/[1+({Y₂-k)/(X₂-h)}*{Y₃-k)/(X₃-h)}]) = tan θ ------------ (1) and

    (k-Y₂)² + (h-X₂)² = d² --------------------------------------------------------------------------------------(2)

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