SOLUTION: What is the legnth of the hypotenuse of a 30-60-90 triangle whose longer leg has a length of 6 radical 3 ft?

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Question 617680: What is the legnth of the hypotenuse of a 30-60-90 triangle whose longer leg has a length of 6 radical 3 ft?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
SHORT ANSWER:
The fastest way to write an answer seems to be invoking trigonometric ratios.
The long leg is opposite the 60%5Eo angle, so
sin%2860%5Eo%29=long_leg%2Fhypotenuse --> hypotenuse=sin%2860%5Eo%29%2Along_leg
long_leg=6%2Asqrt%283%29
and sin%2860%5Eo%29=sqrt%283%29%2F2
Putting it al together


HOW WE KNOW THOSE TRIGONOMETRIC RATIOS:
A 30-60-90 triangle is half of an equilateral triangle:

AB=AD, so a=BC=AD=2%2AAB=2c (The hypotenuse is twice as long as the short leg).
Using the Pythagorean theorem (b%5E2%2Bc%5E2=a%5E2), we get
b%5E2%2Bc%5E2=%282c%29%5E2 --> b%5E2%2Bc%5E2=4c%5E2 --> b%5E2=4c%5E2-c%5E2 --> b%5E2=3c%5E2 --> b=sqrt%283c%5E2%29 --> b=sqrt%283%29%2Ac
So the trigonometric ratios in a 30-60-90 triangle are
sin%2830%5Eo%29=cos%2860%5Eo%29=c%2Fa=c%2F%282c%29=1%2F2
cos%2830%5Eo%29=sin%2860%5Eo%29=b%2Fa=%28sqrt%283%29%2Ac%29%2F%282%2Ac%29=sqrt%283%29%2F2

cot%2830%5Eo%29=+tan%2860%5Eo%29=b%2Fc=sqrt%283%29%2Ac%2Fc=sqrt%283%29