To determine the length of the shortest leg in the given right triangle, we need to utilize properties of a 30-60-90 triangle.
A 30-60-90 triangle has side lengths that are consistent based on the following ratios:
- The length of the shorter leg is [tex]\(a\)[/tex]
- The length of the longer leg (adjacent to the 60-degree angle) is [tex]\(a \sqrt{3}\)[/tex]
- The hypotenuse is [tex]\(2a\)[/tex]
Given that the length of the longer leg is 15, we can set up an equation based on the known ratios:
[tex]\[ \text{longer leg} = a \sqrt{3} \][/tex]
[tex]\[ 15 = a \sqrt{3} \][/tex]
To solve for [tex]\(a\)[/tex], we divide both sides of the equation by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ a = \frac{15}{\sqrt{3}} \][/tex]
Rationalizing the denominator, we multiply the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ a = \frac{15 \sqrt{3}}{3} \][/tex]
[tex]\[ a = 5 \sqrt{3} \][/tex]
Thus, the length of the shortest leg is [tex]\(5 \sqrt{3}\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{5 \sqrt{3}} \][/tex]
