A triangle with angle measures 30°, 60°, 90° has a perimeter of 18+6√3. What is the length of the longest side of the triangle?

 

Generally speaking, the sides of a 30°-60°-90° triangle look like this:

Sum those up, and you get a perimeter of x+2x+x\sqrt{3}=3x+x\sqrt{3}. The triangle in this question has a perimeter of 18+6\sqrt{3}. Set those equal to solve for x:

3x+x\sqrt{3}=18+6\sqrt{3}
x\left(3+\sqrt{3}\right)=6\left(3+\sqrt{3}\right)
x=6

BE CAREFUL THOUGH! The question didn’t ask for x, it asked for the length of the longest side! The longest side (AKA hypotenuse) is 2x=2(6)=12.

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