Certainly! When we analyze this triangle, we see that it is a special type of triangle known as a 30-60-90 triangle. In a 30-60-90 triangle, the sides have a specific ratio: the length of the shortest side (opposite the 30-degree angle) is half the length of the hypotenuse, and the length of the side opposite the 60-degree angle is the shortest side multiplied by [tex]\( \sqrt{3} \)[/tex].
Given:
- The shortest side is [tex]\( 3 \sqrt{3} \)[/tex] inches.
- One angle is [tex]\( 60^{\circ} \)[/tex].
### Step-by-Step Solution:
1. Identify the sides of the triangle:
- The shortest side is opposite the [tex]\( 30^{\circ} \)[/tex] angle.
- The side opposite the [tex]\( 60^{\circ} \)[/tex] angle will be the shortest side multiplied by [tex]\( \sqrt{3} \)[/tex].
- The hypotenuse is twice the length of the shortest side.
2. Calculate the hypotenuse:
- The hypotenuse [tex]\( h \)[/tex] can be found using the relationship:
[tex]\[
h = 2 \times (\text{shortest side})
\][/tex]
- Substituting the given shortest side:
[tex]\[
h = 2 \times 3 \sqrt{3}
\][/tex]
3. Simplify the hypotenuse:
[tex]\[
h = 6 \sqrt{3}
\][/tex]
### Conclusion:
The length of the hypotenuse of the triangle is [tex]\( 6 \sqrt{3} \)[/tex] inches.
Therefore, the correct answer is:
D. [tex]\( 6 \sqrt{3} \)[/tex]
