The length of the hypotenuse of a 30-60-90 triangle is 18. what is the perimeter?

The answer is 27 + 9√3.

Let c be a hypotenuse of a triangle and a and b two legs.

c = 18

In a 30°-60°-90° triangle, the hypotenuse (c) is twice the length of the shorter leg (a):
c = 2a   ⇒ a =  c ÷ 2 = 18 ÷ 2 = 9

In a 30°-60°-90° triangle, the longer leg is equal to the shorter leg multiplied by √3:
b = √3a = √3 · 9 = 9√3

Now we have the length of all three sides:
a = 9
b = 9√3 = √6² · √3 = √36 · √3 = √(36 · 3) = √108
c = 18

So, the perimeter (P) of the triangle is:
P = a + b + c = 9 + 9√3 + 18 = 27 + 9√3