SOLUTION: What is the proof for Theorem 7.9 30° -60° -90° Triangle: In a 30° -60° -90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as l

Algebra ->  Geometry-proofs -> SOLUTION: What is the proof for Theorem 7.9 30° -60° -90° Triangle: In a 30° -60° -90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as l      Log On


   

Question 407148: What is the proof for Theorem 7.9 30° -60° -90° Triangle: In a 30° -60° -90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Use the Law of Sines:

a%2F%28sin%2830%29%29+=+b%2F%28sin%2860%29%29+=+c%2F%28sin%2890%29%29

Substitute sin%2830%29+=+1%2F2, sin%2860%29+=+sqrt%283%29%2F2, sin%2890%29+=+1 to get

a%2F%281%2F2%29+=+b%2F%28sqrt%283%29%2F2%29+=+c

2a+=+2b%2Fsqrt%283%29+=+c. Here, we get c = 2a, and b+=+a%2Asqrt%283%29 as desired.

Interesting fact: The law of sines can be extended and it actually says
a%2F%28sin%28alpha%29%29+=+b%2F%28sin%28beta%29%29+=+c%2F%28sin%28gamma%29%29+=+2R
where alpha, beta, gamma are the angles opposite a,b,c and R is the circumradius.