To determine the length of the shorter leg in a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle where the longer leg is [tex]\(16\sqrt{3}\)[/tex], follow these steps:
1. Understanding the Triangle Ratios:
A [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle has side lengths in a specific ratio:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (shorter leg) is denoted as [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (longer leg) is [tex]\(\sqrt{3} \cdot x\)[/tex].
- The hypotenuse is [tex]\(2x\)[/tex].
2. Given Information:
We are given the length of the longer leg (opposite the [tex]\(60^\circ\)[/tex] angle), which is [tex]\(16\sqrt{3}\)[/tex].
3. Relating the Longer Leg to the Shorter Leg:
The longer leg ([tex]\(16\sqrt{3}\)[/tex]) equals [tex]\(\sqrt{3} \times x\)[/tex], where [tex]\(x\)[/tex] is the length of the shorter leg.
[tex]\[
\text{Longer leg} = \sqrt{3} \times \text{Shorter leg (}x\text{)}
\][/tex]
[tex]\[
16\sqrt{3} = \sqrt{3} \times x
\][/tex]
4. Solving for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex] (the shorter leg), divide both sides of the equation by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
x = \frac{16\sqrt{3}}{\sqrt{3}}
\][/tex]
[tex]\[
x = 16
\][/tex]
Therefore, the length of the shorter leg is [tex]\(16\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{16}
\][/tex]
