To determine the length of the hypotenuse of a right triangle where the shortest side measures [tex]\(3 \sqrt{3}\)[/tex] inches and one angle measures [tex]\(60^\circ\)[/tex], we can use the properties of special right triangles, specifically the 30-60-90 triangle.
A 30-60-90 triangle has sides in a specific ratio, which is:
- The side opposite the 30° angle is [tex]\( x \)[/tex].
- The side opposite the 60° angle is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse (opposite the 90° angle) is [tex]\( 2x \)[/tex].
In this problem, the shortest side opposite the 30° angle measures [tex]\(3 \sqrt{3}\)[/tex] inches. Therefore, we can set:
[tex]\[ x = 3 \sqrt{3} \][/tex]
To find the hypotenuse, which is [tex]\( 2x \)[/tex], we use the value of [tex]\( x \)[/tex]:
[tex]\[ \text{Hypotenuse} = 2x = 2 \cdot 3 \sqrt{3} \][/tex]
Simplify the expression:
[tex]\[ \text{Hypotenuse} = 6 \][/tex]
Therefore, the length of the hypotenuse is [tex]\( 6 \)[/tex] inches.
The correct answer is D. 6.
