I thought myself that you could rotate one arbitrary side, which is connected to the other side, over 90 degrees to the left and 90 degrees to the right, so basically there are infinitely many triangles possible.
Am I right?
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2Yes, that is correct. – lulu Oct 24 '19 at 14:53
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1Yep. You can (for example) uniquely define the triangle by specifying the angle between the two sides, so without this information, there are infinitely many choices for the angle between the sides. – John Doe Oct 24 '19 at 14:54
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If you have two sides $a, b$, the third side will be in this interval $|a-b|<c<a+b$ so yes, infinitely many. – Vasili Oct 24 '19 at 14:55
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Allright, thanks guys – mathomato Oct 24 '19 at 14:55
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To define a triangle, you need three bits of information including at least one side length. Since you only have two, there are infinitely many triangles that satisfy your criteria. – Andrew Chin Oct 24 '19 at 16:00
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I agree with the other answers but would like to present a geometric argument which is very similar to the analysis already indicated by the OP.
Assume that two of the sides have a length of $x$ and $r$, where $x$ and $r$ are both positive. Assume that one of the vertices of the triangle will have Cartesian coordinates $(x,0).$ Further assume that the origin [i.e. Cartesian coordinates $(0,0)$] will be a 2nd vertex of the triangle.
Now, consider a circle centered at the origin, of radius $r$. You can select any point on this circle as the third vertex of the triangle [except for the points $(r,0)$ and $(-r,0)$].
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