To determine the length of the hypotenuse in a right triangle where the shortest side measures [tex]\(3 \sqrt{3}\)[/tex] inches and one angle is [tex]\(60^\circ\)[/tex], consider the properties of a 30-60-90 triangle.
In a 30-60-90 triangle:
- The ratio of the sides is [tex]\(1\)[/tex] : [tex]\(\sqrt{3}\)[/tex] : [tex]\(2\)[/tex].
- The shortest side (opposite the [tex]\(30^\circ\)[/tex] angle) corresponds to the [tex]\(1\)[/tex] in the ratio.
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex] times the length of the shortest side.
- The hypotenuse is [tex]\(2\)[/tex] times the length of the shortest side.
Given that the shortest side is [tex]\(3\sqrt{3}\)[/tex]:
- The hypotenuse is [tex]\(2\)[/tex] times this shortest side.
So, the hypotenuse is:
[tex]\[ 2 \times 3\sqrt{3} = 6\sqrt{3} \][/tex]
Thus, the length of the hypotenuse is [tex]\(6\sqrt{3}\)[/tex] inches.
Therefore, the correct answer is:
A. [tex]\(6 \sqrt{3}\)[/tex]
