SOLUTION: The angles of the triangle are 30°-30°-120°. Two of the sides lengths are 2 times the square root of 3. What is the length of the third side?

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Question 310449: The angles of the triangle are 30°-30°-120°. Two of the sides lengths are 2 times the square root of 3. What is the length of the third side?
Found 3 solutions by solver91311, Edwin McCravy, Kartik-The Math specialist:
Answer by solver91311(24713) About Me  (Show Source):
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The hypotenuse is the longest side of a right triangle. It is always the side opposite the right angle. Since your 30-30-120 triangle doesn't have a right angle, it is not a right triangle, therefore it does not have a hypotenuse.

Having said that, if you want to know the measure of the longest side of your triangle, even though it isn't called a hypotenuse, use the Law of Cosines.

Let c represent the measure of the longest side, and let a and b represent the measures of the other two equal length sides. Let C represent the measure of the largest angle.

Then



which, for our purposes will be more conveniently written:



Plugging in the numbers given:



You can do your own arithmetic. Hint: and


John

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutor assumed you have studied the law of cosines.
I am guessing that you haven't. I think you've only had the
Pythagorean theorem.  Maybe you haven't even gotten to
trigonometry.  Am I right? If so then do it this way:

The angles of the triangle are 30°-30°-120°. Two of the sides lengths are 2
times the square root of 3. What is the length of the third side?



The triangle is isosceles because two sides and two angles have the
same measure.  So draw a perpendicular to the base, which also
bisects both the third side as well as the 120° vertex angle. like this:



It bisects the 120° into two 60° angles like this:



Let each of the two halves of the third side be x:



Now for the right triangle on the left:



Since this is a 30°-60°-90° right triangle, we know that
the shorter leg (the green side) is one-half of the hypotenuse.
So we can label it one-half of  2sqrt%283%29, which is 
sqrt%283%29, like this:


 
 Now we can use the Pythagorean theorem to find the longer leg:

x%5E2%2B%28sqrt%283%29%29%5E2=%282sqrt%283%29%29%5E2

x%5E2%2B3=2%5E2%2A%28sqrt%283%29%29%5E2

x%5E2%2B3=4%2A3

x%5E2%2B3=12

x%5E2=12-3

x%5E2=9

x=sqrt%289%29

x=3

And so the right triangle has longer leg of 3



Now we put the original triangle back together like this:



And we end up with



Edwin

Answer by Kartik-The Math specialist(4) About Me  (Show Source):
You can put this solution on YOUR website!
See I have created a theorem which states that if in a triangle the angle between two equal sides is 120 then the 3rd side is ROOT 3 * Equal sides........
SO THIRD SIDE IN THIS QUESTION IS EQUAL TO ROOT 3* 2*ROOT 3 = 6.
If u want to know the proof of my theorem.
Contact me........that is 9471316216.....BCOZ I M THE BEST