math
posted by jay on .
Help me I am lost? I do not even know where to start:
The shorter leg of a 30°–60°–90° triangle is 12. What are the lengths of the longer leg and the hypotenuse, to the nearest tenth?

First, draw your triangle so you can see what you're doing. The short side, 12, will be opposite the small angle, 30.
We know that the sin of an angle = opposite over hypotenuse. The angle is 30, and we can look up sin(30)=0.5.
So opposite (which we know is 12) over hypotenuse = 1/2.
12/hypotenuse = 1/2
hypotenuse = 24.
Now we can get the final, adjacent, side in either of two ways:
1. Pythagoras: 12^2 + adjacent^2 = 24^2
or
2. use the definition of cos = adjacent over hypotenuse. We can look up cos(30) = sqrt(3)/2 or .8660. So:
adjacent/24 = .8660
and mutiply across from there. 
The sides of the 306090 triangel are in the ratio of 1:√3:2
your triangle is 12:a:b
so 1/12 = √3/a
a = 12√3
and 1/12 = 2/b
b = 24
the other two sides are 12√3 and 24