What is the length of the hypotenuse of a 30-60-90 triangle with a longer leg of length 16m?
To find the length of the hypotenuse of a 30-60-90 triangle, you can use the properties of this special type of right triangle. In a 30-60-90 triangle, the lengths of the sides are related by a specific ratio.
The ratio for a 30-60-90 triangle is:
- The length of the shorter leg is always half the length of the hypotenuse.
- The length of the longer leg is equal to √3 times the length of the shorter leg.
Given that the longer leg of the triangle is 16 m, we can use the ratio to find the length of the hypotenuse.
Step 1: Find the length of the shorter leg.
Since the longer leg is 16 m and it is equal to √3 times the length of the shorter leg, we can set up the equation:
16 = √3 * x, where x is the length of the shorter leg.
To solve this equation, divide both sides by √3:
16/√3 = x.
Step 2: Simplify the expression for the length of the shorter leg.
To simplify 16/√3, we can multiply both the numerator and denominator by √3:
(16/√3) * (√3/√3) = (16√3) / 3.
So, the length of the shorter leg is (16√3) / 3 meters.
Step 3: Find the length of the hypotenuse.
Using the ratio, the length of the hypotenuse is twice the length of the shorter leg:
Hypotenuse = 2 * (16√3) / 3.
To simplify this expression, multiply the numerator and denominator by 2:
Hypotenuse = (32√3) / 3.
Therefore, the length of the hypotenuse of the 30-60-90 triangle is (32√3) / 3 meters.
2 / √3 = h / 16
h = 32 / √3 = 32√3 / 3