To solve this problem, we start by understanding the properties of a right triangle with one angle measuring [tex]\( 60^\circ \)[/tex].
Given:
- The shortest side (opposite the [tex]\( 30^\circ \)[/tex] angle) is [tex]\( 3 \sqrt{3} \)[/tex] inches.
- One angle of the triangle is [tex]\( 60^\circ \)[/tex].
In a right triangle, the primary trigonometric relationships can be helpful. Here, we can use the sine function to find the hypotenuse. For a given angle in a right triangle, the sine of the angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Symbolically, for angle [tex]\( \theta \)[/tex],
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
Given:
- Opposite side [tex]\( = 3 \sqrt{3} \)[/tex] inches
- Angle [tex]\( = 60^\circ \)[/tex]
We need to find the hypotenuse [tex]\( h \)[/tex]. Rearranging the sine formula to solve for the hypotenuse gives us:
[tex]\[ \sin(60^\circ) = \frac{3 \sqrt{3}}{h} \][/tex]
The sine of [tex]\( 60^\circ \)[/tex] is:
[tex]\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \][/tex]
Substitute this value into the equation:
[tex]\[ \frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{h} \][/tex]
Next, solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3 \sqrt{3}}{\frac{\sqrt{3}}{2}} = 3 \sqrt{3} \times \frac{2}{\sqrt{3}} \][/tex]
Simplify the expression:
[tex]\[ h = 3 \times 2 \][/tex]
[tex]\[ h = 6 \][/tex]
Therefore, the length of the hypotenuse is 6 inches.
Thus, the correct answer is:
A. 6
