I have two points in an (x,y) coordinate system, the only information i have is the distance of these points from the origin and the distance between these two points. I need to find the coordinates of these two points. As these points form a triangle so i use the properties of triangles to find these coordinates. Using the sides length information i can find out the three angles inside. But still i haven't been able to find the coordinates. The attached pictures can help to visualize the problem.
The solution is not unique: for every solution, if you turn the whole picture around zero for some fixed angle, you obtain another solution. So, you may assume that the point $(x_1, y_1)$ equals $(5, 0)$ and obtain the remaining solutions by the above argument. For this particular case you obtain two equations for unknown point $(x_2, y_2)$: $(x_2)^2 + (y_2)^2 = 16$ (the distance from the point to zero equals 4) and $(x_2 - 5)^2 + (y_2)^2 = 36$ (the distance from the point to $(5, 0)$ equals 6). Solving this system of 2 equations with 2 unknowns you obtain the solution (or more precisely, two solutions, above and below the x axes).
Vladimir Kadets Thanks for the help you have provided. Can you give me some suggestion, what necessary information must be included to find a unique solution. The lesser the initial information is, the better its going to be in my work.
Sohaib Bin Altaf Khattak, for example, it is sufficient to know what is the angle between the x-axes and the vector that goes from (0, 0) to (x_1, y_1), and the information whether the enumeration (0, 0), (x_1, y_1), (x_2, y_2) is clockwise or anticlockwise.
Vladimir Kadets In this problem if I assume, I have the coordinates of two points, one is on the origin (0,0) and the other point is also on any of the axis with coordinates as (A,0) or (0,B), then its easy to find a generalized equation for the third coordinate of the triangle. But is it possible to find the (x,y) of the third coordinate if the coordinates of the second point are not on the axis (A,B) instead of (A,0) or (0,B). I am trying to solve it but so far I am unable to find an answer. If you could say anything about it, it would be really helpful.
Sohaib Bin Altaf Khattak You can deduce the general case from the particular one with the help of rotation of axes: https://en.wikipedia.org/wiki/Rotation_of_axes
Vladimir Kadets Thank you very much, I will read and see how i can use it.
Clearly given the side lengths of the triangle OM1M2 and the position of one point (namely:the origin O) the coordinates of M1, M2, and even if we are given the coordinates of M1 or M2 there are two possible positions of the other point.
Omran Kouba If I keep a limitation, that the third point must be in a particular quadrant, then it will be easy to find a unique solution?
Sohaib Bin Altaf Khattak Yes, if you fix O and M1 and you request that the triangle is positively oriented for example then the third point is uniquely determined.
Omran Kouba I have been trying to do it with these values, to form a generalized equation, where i can find the (x,y) for the third point, but i am unable to find an answer. when i make the first pt as (0,0) and second pt as either (A,0) or (B,0), I can derive an equation for x and y, but one the second pt is (A, B) both are nonzero, what should i do?
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