Help me I am lost? I do not even know where to start:

Question

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?

Answer 1

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.

Answer 2

The sides of the 30-60-90 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