To solve for the lengths of the sides in a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, we can use a well-known property of this type of triangle: the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex].
Given:
- Hypotenuse = 30
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- The hypotenuse is twice the length of the shorter leg.
- The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Step 1: Find the length of the shorter leg.
Since the hypotenuse is twice the length of the shorter leg:
[tex]\[
\text{Shorter leg} = \frac{\text{Hypotenuse}}{2} = \frac{30}{2} = 15
\][/tex]
Step 2: Find the length of the longer leg.
The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg:
[tex]\[
\text{Longer leg} = \text{Shorter leg} \times \sqrt{3} = 15 \times \sqrt{3}
\][/tex]
Simplifying the expression for the longer leg, we get:
[tex]\[
\text{Longer leg} = 15\sqrt{3}
\][/tex]
Therefore, the length of the longer leg is [tex]\(15\sqrt{3}\)[/tex], which corresponds to option D.
So, the best answer from the choices provided is:
D. [tex]\(15\sqrt{3}\)[/tex]
