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3. The longer leg of a [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangle is [tex]$16 \sqrt{3}$[/tex]. How long is the shorter leg?
A. 16 B. [tex][tex]$8 \sqrt{3}$[/tex][/tex] C. [tex]$16 \sqrt{2}$[/tex] D. 32
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]
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