# What is the name of this geometry theorem?

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• barryj
barryj
In geometry, there is a theorem pertaining to whether given an angle, side, and side gives 0, 1, or 2 triangles. See figure. In the figure, if x = 10sin(30) then there is exactly 1 triangle, if x > 10sin(30) then 2 triangles if x < 10sin(30) then no triangles. I think this has a theorem name or something that I can look up in a tggext book.

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I am not sure I get you. Your drawing is no triangle case, right ? Could you draw "2 triangles" case to confirm I get you properly.

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The condition for three sides, A, B, and C to make a triangle is that the sum of any two sides must be greater than the third side. Given an angle and two sides the question is will these form a triangle. There are three conditions depending on how long the "hanging" side is. In my diagram, the hanging side is x the other side is 10, and the angle is 30 deg. If x = 10sin(30) then there will be one right triangle. If x < 10sin(30) then there is no triangle, and if x > 10sin(30) then there will be two triangles. I guess there is not a theorem here, just a problem.

Thanks for explanation. Law of cosine
$$c^2=a^2+b^2-2ab \cos \gamma$$
Regarding this as quadratic equation of b
$$b^2-2a \cos \gamma \ b + a^2-c^2 = 0$$
$$D/4= (a \cos \gamma)^2 - a^2 + c^2 = ( c- a \sin \gamma )( c + a \sin \gamma)$$
##a \sin \gamma < c## : b has two real solutions with ##\alpha_1<\frac{\pi}{2}<\alpha_2##
##a \sin \gamma = c## : b has one real solution with ##\alpha=\frac{\pi}{2}##
##a \sin \gamma > c## : b has no real solution

I do not know whether this feature has a specific name or not.

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mathwonk and Lnewqban
barryj said:
In geometry, there is a theorem pertaining to whether given an angle, side, and side gives 0, 1, or 2 triangles. See figure. In the figure, if x = 10sin(30) then there is exactly 1 triangle, if x > 10sin(30) then 2 triangles if x < 10sin(30) then no triangles. I think this has a theorem name or something that I can look up in a tggext book.
I think it is called the law of Sines, since you are given SSA conditions on the triangle. This case is called the ambiguous case because there could be 0,1, 2 triangles.

cbarker1

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