Question

The length of the hypotenuse of a 30°-60°-90° triangle is 38. What is the perimeter?

1. 19+19sqrt(3)
2. 57+19sqrt(3)
3. 57+19sqrt(2)
4. 38+38sprt(2)​

Answer 1

Explanation:

In a 30°-60°-90° triangle, the sides are in the ratio of 1:√3:2.

Let x be the length of the shorter leg (opposite the 30° angle). Then, the longer leg (opposite the 60° angle) is x√3, and the hypotenuse (opposite the 90° angle) is 2x.

We are given that the hypotenuse is 38, so:

2x = 38

x = 19

Therefore, the shorter leg is 19, and the longer leg is 19√3.

The perimeter of the triangle is the sum of the lengths of all three sides:

19 + 19√3 + 38 = 57 + 19√3

So the answer is (2) 57+19sqrt(3).

Answer 2

We know that in a 30°-60°-90° triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is $\frac{\sqrt{3}}{2}$ times the length of the hypotenuse.

Therefore, the length of the side opposite the 30° angle is $\frac{1}{2}(38) = 19$, and the length of the side opposite the 60° angle is $\frac{\sqrt{3}}{2}(38) = 19\sqrt{3}$.

To find the perimeter, we add up the lengths of all three sides:

Perimeter = 19 + 19$\sqrt{3}$ + 38 = 57 + 19$\sqrt{3}$

Therefore, the answer is option 2, 57 + 19$\sqrt{3}$.