Question

The length of the hypotenuse of a 30°-60°-90° triangle is 12. Find the perimeter.

Answer 1

The answer is 18 + 6√3 or 18 + √108.

In a 30°-60°-90° triangle, the hypotenuse (c) is twice the length of the shorter leg (a):
c = 2a   ⇒ a =  c ÷ 2 = 12 ÷ 2 = 6

In a 30°-60°-90° triangle, the longer leg is equal to the shorter leg multiplied by √3:
b = √3a = √3 · 6

Now we have the length of all three sides:
a = 6
b = 6√3 = √6² · √3 = √36 · √3 = √(36 · 3) = √108
c = 12

So, the perimeter (P) of the triangle is:
P = a + b + c = 6 + 6√3 + 12 = 18 + 6√3 = 18 + √108