To find the length of the hypotenuse in a right triangle where the shortest side is given and one of the angles is [tex]\(60^\circ\)[/tex], we can use the properties of a 30-60-90 triangle. Here are the steps:
1. Identify the properties of a 30-60-90 triangle:
- In a 30-60-90 triangle, the sides have a very specific ratio:
- The side opposite the 30-degree angle is the shortest side.
- The side opposite the 60-degree angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side.
- The hypotenuse is twice the shortest side.
2. Given information:
- The shortest side (opposite the 30-degree angle) is [tex]\(3\sqrt{3}\)[/tex] inches.
- We need to find the hypotenuse.
3. Relate the hypotenuse to the shortest side:
- According to the properties of the 30-60-90 triangle, the hypotenuse is twice the length of the shortest side.
4. Calculate the hypotenuse:
- The hypotenuse = [tex]\(2 \times \text{shortest side}\)[/tex]
- The shortest side is [tex]\(3\sqrt{3}\)[/tex] inches.
- Therefore, the hypotenuse = [tex]\(2 \times 3\sqrt{3} = 6\sqrt{3}\)[/tex] inches.
Thus, the length of the hypotenuse is [tex]\(6\sqrt{3}\)[/tex] inches.
The correct answer is:
A. [tex]\(6\sqrt{3}\)[/tex]
