To find the length of the hypotenuse in the right triangle given the shortest side and one of the angles, follow these steps:
1. Identify the type of right triangle:
The triangle described has a shortest side and a [tex]$60^{\circ}$[/tex] angle. This is a 30-60-90 right triangle, where the angles are [tex]$30^{\circ}$[/tex], [tex]$60^{\circ}$[/tex], and [tex]$90^{\circ}$[/tex].
2. Understand the side ratios of a 30-60-90 right triangle:
In this type of triangle, the sides are in a specific ratio:
- The side opposite the [tex]$30^{\circ}$[/tex] angle (shortest side) is [tex]\( x \)[/tex].
- The side opposite the [tex]$60^{\circ}$[/tex] angle (longer leg) is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
3. Given:
The shortest side is [tex]\( 3 \sqrt{3} \)[/tex] inches. This side is opposite the [tex]$30^{\circ}$[/tex] angle.
4. Set up the ratio and solve for the hypotenuse:
- The shortest side (opposite the [tex]$30^{\circ}$[/tex] angle) is [tex]\( x \)[/tex], so [tex]\( x = 3 \sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
5. Calculate the hypotenuse:
[tex]\[
\text{Hypotenuse} = 2x = 2 \cdot (3\sqrt{3}) = 6\sqrt{3}
\][/tex]
So, we have calculated that the length of the hypotenuse is equal to [tex]\( 6 \sqrt{3} \)[/tex] inches.
6. Check the options:
A. [tex]\( 6 \sqrt{2} \)[/tex]
B. 6
C. 3
D. [tex]\( 6 \sqrt{3} \)[/tex]
The correct answer is:
[tex]\[ \boxed{6} \][/tex]
