To determine the length of the hypotenuse of a right triangle when one angle and one side are given, we can use trigonometric relationships. Here’s the step-by-step process:
1. Identify the given information:
- The shortest side of the right triangle is [tex]\( 3 \sqrt{3} \)[/tex] inches. This side is opposite the [tex]\( 60^\circ \)[/tex] angle.
- One angle of the triangle measures [tex]\( 60^\circ \)[/tex]. Since it is a right triangle, the other non-right angle is [tex]\( 30^\circ \)[/tex].
2. Use trigonometric relationships:
For a right triangle:
- [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex]
We need to find the hypotenuse [tex]\( c \)[/tex]. Let the shortest side ([tex]\( 3 \sqrt{3} \)[/tex]) be the opposite side for the [tex]\( 60^\circ \)[/tex] angle.
3. Set up the equation using the sine function:
[tex]\[
\sin(60^\circ) = \frac{3 \sqrt{3}}{c}
\][/tex]
From trigonometric tables or unit circle, we know:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
4. Solve for the hypotenuse [tex]\( c \)[/tex]:
[tex]\[
\frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{c}
\][/tex]
Cross-multiplying to solve for [tex]\( c \)[/tex]:
[tex]\[
c \cdot \frac{\sqrt{3}}{2} = 3 \sqrt{3}
\][/tex]
Multiply both sides by 2:
[tex]\[
c \sqrt{3} = 2 \cdot 3 \sqrt{3}
\][/tex]
[tex]\[
c \sqrt{3} = 6 \sqrt{3}
\][/tex]
Divide both sides by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[
c = \frac{6 \sqrt{3}}{\sqrt{3}}
\][/tex]
[tex]\[
c = 6
\][/tex]
Thus, the length of the hypotenuse of the triangle is [tex]\( \boxed{6} \)[/tex] inches.
