To solve this problem, let's analyze the characteristics of a right triangle with a [tex]\(60^\circ\)[/tex] angle.
In a right-angled triangle with angles of [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex], the sides are always in a specific ratio: [tex]\(1 : \sqrt{3} : 2\)[/tex].
- The side opposite to the [tex]\(30^\circ\)[/tex] angle (the shortest side) is [tex]\(a\)[/tex].
- The side opposite to the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\(a\sqrt{3}\)[/tex].
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) is [tex]\(2a\)[/tex].
Given in the problem:
- The shortest side measures [tex]\(3\sqrt{3}\)[/tex] inches.
- We are asked to find the length of the hypotenuse.
From the given information, let's define:
- The shortest side [tex]\(a\)[/tex] equals [tex]\(3\sqrt{3}\)[/tex].
Using the given ratio for sides in a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
[tex]\[ a\text{ (short side)} : a\sqrt{3}\text{ (longer leg)} : 2a\text{ (hypotenuse)} \][/tex]
Since the shortest side given is [tex]\(a = 3\sqrt{3}\)[/tex]:
- The hypotenuse will be [tex]\(2a\)[/tex].
Therefore, we calculate the hypotenuse as follows:
[tex]\[ 2a = 2 \times (3\sqrt{3}) = 6\sqrt{3} / \sqrt{3} = 6 \][/tex]
Thus, the length of the hypotenuse is [tex]\(6\)[/tex] inches.
The correct answer is:
C. 6
