I What is the name of this geometry theorem?

barryj
Messages
856
Reaction score
51
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.

img439.jpg
 
Last edited by a moderator:
Mathematics news on Phys.org
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.
 
Last edited by a moderator:
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.
 
Last edited:
  • Like
Likes 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
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Replies
38
Views
3K
Replies
3
Views
2K
Replies
1
Views
2K
Replies
59
Views
1K
Replies
9
Views
1K
Replies
11
Views
2K
Replies
2
Views
1K
Back
Top