Sure! Let's solve the problem step-by-step.
A [tex]\(30^{\circ}-60^{\circ}-90^{\circ}\)[/tex] triangle has specific side length ratios:
- The side opposite the [tex]\(30^{\circ}\)[/tex] angle (short leg) is [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^{\circ}\)[/tex] angle (long leg) is [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2x\)[/tex].
Given that the hypotenuse is 4, we can find [tex]\(x\)[/tex] by setting up the equation for the hypotenuse:
[tex]\[
2x = 4
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4}{2} = 2
\][/tex]
Now, we need to find the length of the longer leg, which is given by [tex]\(x\sqrt{3}\)[/tex]:
[tex]\[
\text{Longer leg} = x\sqrt{3} = 2\sqrt{3}
\][/tex]
So, the length of the longer leg is [tex]\(2\sqrt{3}\)[/tex].
Therefore, the correct answer is:
[tex]\[
2\sqrt{3}
\][/tex]
