To solve this problem, it's important to note that we're dealing with a 30-60-90 right triangle, which has some special properties. In a 30-60-90 triangle, the relationships between the lengths of the sides are always consistent. These relationships are:
1. The side opposite the 30-degree angle (the shortest side) has a length of [tex]\( x \)[/tex].
2. The side opposite the 60-degree angle (the medium side) is [tex]\( x\sqrt{3} \)[/tex].
3. The hypotenuse (the side opposite the 90-degree angle) is [tex]\( 2x \)[/tex].
In this problem, we are given that the shortest side (opposite the 30-degree angle) measures [tex]\( 3\sqrt{3} \)[/tex] inches. According to the special properties of a 30-60-90 triangle:
[tex]\[ x = 3\sqrt{3} \][/tex]
The hypotenuse in a 30-60-90 triangle is twice the length of the shortest side. Therefore:
[tex]\[ \text{Hypotenuse} = 2x = 2(3\sqrt{3}) \][/tex]
By calculating the above expression, we get:
[tex]\[ \text{Hypotenuse} = 6\sqrt{3} \][/tex]
So, the length of the hypotenuse is [tex]\( 6\sqrt{3} \)[/tex] inches.
Thus, the correct answer is:
D. [tex]\( 6\sqrt{3} \)[/tex]
