QR Trig Question

[Meat Gyver]

Hi, I was hoping that someone could shed some light on a question I ran into on the free practice DAT bootcamp QR exam. (It's question 29, but I'm not sure if the questions are randomized)

The question asks "In the figure below, given that the side length "X" is equal to 5, what is a possible value for the side length of "Y"? The figure is not drawn to scale."

(attached is a [poor quality] image of the question).

The answer choices are: 3, 5, 7, 9, 11

Correct Answer: 9

My answer choice: 9.8

This is the explanation on the website:

We know that X is 5, and that tan(30) = x/y. Tan(30) is approximately 0.57, or . This can be found through a unit circle using sin(30)/cos(30). I suggest drawing out the unit circle during your break between the first and second part of the DAT. Thus, 5/y = 0.57. The closest value to this is D, or 9. This is confirmed by 5/11 = 0.56.

Seeing by the image, I took a different approach to the question by using the law of sines. I was just curious to know why my approach to answering the question was wrong.

Thanks for your help!

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[bluemoon27]

I dont understand, the tan theta reasoning, is that what you used? Because that can only be used for right triangle. This triangle is not a right triangle.

However, my approach was a bit different where I tried to turn it into a right triangle, but then the angles dont work out. If you put an vertical bisector in the middle of the side where x = 5, so that it cuts the 30 into 15 on each side, but if you do the math, it actually changes the left angle to 75 and not 70. This is really weird! I have never encountered something like this before. Unless Im making a dumb mistake.

But I think that law of sines is the only way to do this.

Can anyone else elaborate? Im sort of lost too.

 

[Meat Gyver]

I'm on the same boat as you. I'm totally lost on the Tan theta explanation. I initially just used the Law of Sines (seeing that it applies to every triangle, not just right triangles) to get an answer of 9.8 (rounded roughly to 10). But, that was right in between 9 and 11, so I was stuck between answer choices. Then I actually also used the method you mentioned on bisecting the 30 degrees, but that just distorts the internal angles. To use Tan(30), like you said, we would have to assume the triangle is a right triangle.

Possible errors: Maybe the lower triangle is supposed to be a 30 (not 40)-60-90 triangle, and the internal angle of the triangle we are interested is 90 degrees as opposed to 80 degrees. However, to solve that, you'd use Sin(30) = 5/y (and not Tan(30). Even despite that the answer would still come out to 10.