Why can I not use an equation using proportions to solve this triangle problem?

Question

It is difficult to see the picture of the problem. The question is "What are the lengths of AC and AB?" What is given is a right triangle, ABC. Angle B is 30 degrees and BC is 7.0 distance.

The way I solved it was by using the properties of a 30 60 90 degree triangle which I learned from a unit circle. The segment opposite the 30 degree angle is 1/2 the distance of the hypotenuse. And the segment opposite the 60 degree angle is square root of 3 times the length of the side opposite 30 degrees. (I think that's correct). I got it right.

Though, I originally tried to solve the problem though using an equation using proportions. I figured: Alright 90 degrees over 7.0 is proportionate to 30 degrees over x. Thus: 90/7 = 30/x and I solved it. However, I did not arrive at the correct answers. Can someone help me understand why this does not work? Thanks, Paige

Answer 1

As long as it's a 30-60-90 triangle, you can always use the proportions. You're thinking about the wrong proportions, however: the lengths are not exactly proportional to the angles. You'll get to that in trig with law of cosine, but, for now, the sides are proportional to each other in the way presented in the image above. Above is a triangle similar to the one you presented (i.e. angles at the same place) with some variables $a$ to describe the proportions. We know that, in your case, the side that's $2a$ is equal to 7. Hence: $$2a=7$$$$a=3.5$$

Now that we know what $a$ is, we can find the lengths easily. AC is the side opposite of the 30-degree angle and is equivalent to $a$. Therefore, it is 3.5. That already eliminates you down to choices C and D. Then we have AB, which is opposite of the 60-degree angle and is equal to $a\sqrt{3}$. This is approximately equal to $6.1$.

The final answer is, then, D.

Answer 2

I just ready your question. First and fore mostly I apologize if my answer isn't in the right syntax, mathematical grammar etc. This is my first stack exchange question. The easiest way I have found to solve these problems is to get the ratio for one leg and work out from there. For instance, the sides of a 30-60-90 triangle are $1, \sqrt{3}, \text{ and } 2$ (2 being the hypotenuse) or multiples of such. In your triangle, there is a hypotenuse of 7. To get the ratio, I divided $7$ by $2$. This gave me $3.5$. Now to get the other sides, I simply multiplied the ratio by the sides of the original 30-60-90 triangle and got the answer, D (i.e. $\sqrt{3} \times 3.5$ = a rounded $6.1$ and $1 \times 3.5 = \text{(duh) } 3.5$ which are answer choices). I don't know if that is what is what you meant about proportions, but it's the best way I've found.