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- Thread starter Tyrion101
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RUber

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If you look at basic geometry texts about triangle congruence, you will find that various combinations of side lengths and interior angles uniquely determine a triangle. If you have 3 peices of information you might have a unique triangle.

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I wasn't aware of the term but thanks!

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I still can't figure out what it is that you are asking.

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Mark44

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I think I do.I still can't figure out what it is that you are asking.

In a problem for which you're asked to "solve" a triangle, you are typically given three pieces of information. If you are given two sides and the included angle, then the length of the third side will be unique. There is just one way that the ends of the two sides can be connected, and this fixes the length of the unknown side.

OTOH, if you are given two sides and an angle that is not the included angle (not the angle formed by the two sides), then it is usually the case that there will be two solutions for the unknown side. This situation often shows up when you use the Law of Sines. One reason for this is that ##\sin(\pi/2 + A) = \sin(\pi/2 - A)##. Both of these angles have the same sine.

The best way to deal with all of these situations is to draw a sketch that uses the given information.

- #6

HallsofIvy

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There are

1) The arc might not cross the second line at all- the radius is too short. There is NO such triangle.

2) The arc might be tangent to the second line. This is the case where there is exactly one triangle- and it is a

3) The arc might cross the second line in two different points. This is the case where there are two such triangles.

Frankly, the simplest way to do determine whether there is no such triangle, one, or two is to actually try to solve the triangle. The standard way to solve this problem is to use the

We can solve that using the quadratic formula:

[tex]\frac{2a cos(C)\pm \sqrt{4a^2 cos^2(C)+ 4a^2- 4c^2}}{2}= a cos(C)\pm\sqrt{a^2 cos^2(C)+ a^2- c^2}[/tex].

Whether there is zero, one, or two solutions to that, depends on the

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