To solve this problem, let's break it down step by step.
1. Identify the Triangle Type: The problem gives us a right triangle with one angle measuring [tex]\(60^{\circ}\)[/tex]. This means we are dealing with a 30-60-90 triangle, which has specific side length ratios.
2. Side Length Ratios of a 30-60-90 Triangle: In a 30-60-90 triangle, the side lengths are in a specific ratio:
- The side opposite the [tex]\(30^{\circ}\)[/tex] angle is the shortest side, often referred to as the "shorter leg."
- The side opposite the [tex]\(60^{\circ}\)[/tex] angle is the "longer leg."
- The side opposite the [tex]\(90^{\circ}\)[/tex] angle (the hypotenuse) is the longest side.
The side lengths are in the ratio: 1 (shorter leg) : [tex]\(\sqrt{3}\)[/tex] (longer leg) : 2 (hypotenuse).
3. Given Side Correspondence: We are given that the shortest side (opposite the [tex]\(30^{\circ}\)[/tex] angle) is [tex]\(3\sqrt{3}\)[/tex] inches. This means that the longer leg (opposite the [tex]\(60^{\circ}\)[/tex] angle) is [tex]\(\sqrt{3}\)[/tex] times that of the shortest side, and the hypotenuse is twice the shortest side.
4. Determine Hypotenuse: Since we know that the hypotenuse is twice the length of the shortest side in a 30-60-90 triangle:
[tex]\[
\text{Hypotenuse} = 2 \times (\text{shorter leg})
\][/tex]
Given the shorter leg is [tex]\(3\sqrt{3}\)[/tex]:
[tex]\[
\text{Hypotenuse} = 2 \times 3\sqrt{3} = 6\sqrt{3}
\][/tex]
However, it appears there may have been a misunderstanding earlier. The correct given side was the longer leg, not the shorter leg. If we instead treat the side [tex]\(3\sqrt{3}\)[/tex] as the longer leg (opposite [tex]\(60^\circ\)[/tex]), the following ratio applies:
[tex]\[
\text{longer leg} = \sqrt{3} \times (\text{shorter leg})
\][/tex]
Thus:
[tex]\[
3\sqrt{3} = \sqrt{3} \times (\text{shorter leg})
\][/tex]
Solving for the shorter leg:
[tex]\[
\text{shorter leg} = 3
\][/tex]
The hypotenuse, being twice the shorter leg, would then be calculated as:
[tex]\[
\text{Hypotenuse} = 2 \times (\text{shorter leg})
\][/tex]
Given the shorter leg is [tex]\(3\)[/tex]:
[tex]\[
\text{Hypotenuse} = 2 \times 3 = 6
\][/tex]
5. Final Answer: The length of the hypotenuse of the right triangle is [tex]\(6\)[/tex] inches.
Thus, the correct answer is:
B. 6
