To solve for the length of the hypotenuse of the given right triangle, we need to consider the properties of a 30-60-90 triangle. In a 30-60-90 triangle, the sides have a specific ratio: 1 : [tex]\(\sqrt{3}\)[/tex] : 2. The shortest side is opposite the 30-degree angle, the side opposite the 60-degree angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side, and the hypotenuse is twice the shortest side.
Given:
- One angle measures [tex]\(60^{\circ}\)[/tex] which means the triangle has angle measures [tex]\(30^{\circ}\)[/tex] and [tex]\(60^{\circ}\)[/tex], besides the right angle.
- The side opposite the 60-degree angle is given as [tex]\(3 \sqrt{3}\)[/tex] inches.
In the ratio 1 : [tex]\(\sqrt{3}\)[/tex] : 2 for a 30-60-90 triangle:
1. The side opposite the 60-degree angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side. Therefore, let the shortest side be [tex]\(x\)[/tex]. Then,
[tex]\[ x \cdot \sqrt{3} = 3\sqrt{3} \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 3 \][/tex]
2. The hypotenuse is twice the shortest side. Therefore,
[tex]\[ \text{Hypotenuse} = 2 \times 3 = 6 \][/tex]
Thus, the length of the hypotenuse is [tex]\(6\)[/tex] inches.
The correct answer is:
B. 6
