To determine the length of the hypotenuse of the given right triangle, we start with the given information:
1. The shortest side of the right triangle is [tex]\( 3 \sqrt{3} \)[/tex] inches.
2. The triangle has an angle of [tex]\( 60^{\circ} \)[/tex].
Given these facts, it's useful to recognize that we are dealing with a 30-60-90 triangle. In a 30-60-90 triangle, the sides have fixed ratios:
- The side across from the [tex]\( 30^{\circ} \)[/tex] angle is the shortest side and can be denoted as [tex]\(x\)[/tex].
- The side across from the [tex]\( 60^{\circ} \)[/tex] angle is [tex]\(x \sqrt{3}\)[/tex].
- The hypotenuse (across from the right angle) is [tex]\(2x\)[/tex].
Here, the shortest side (across from the [tex]\(30^{\circ}\)[/tex]) is given as [tex]\( 3 \sqrt{3} \)[/tex]. Based on the properties of a 30-60-90 triangle:
- Let's denote the shortest side as [tex]\(x\)[/tex]. Thus, [tex]\( x = 3 \sqrt{3} \)[/tex].
- The hypotenuse is [tex]\(2x\)[/tex].
So, let's calculate the hypotenuse:
[tex]\[
\text{Hypotenuse} = 2 \times 3 \sqrt{3} = 6 \sqrt{3}
\][/tex]
Now we compare this value with the given multiple-choice options:
A. 3
B. [tex]\( 6 \sqrt{2} \)[/tex]
C. [tex]\( 6 \sqrt{3} \)[/tex]
D. 6
The correct answer, based on our calculation, matches option C:
[tex]\[
\boxed{6 \sqrt{3}}
\][/tex]
