Answer :
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the sides are in a well-defined ratio which you need to know. This type of triangle has sides in the ratio:
[tex]\[ 1 : \sqrt{3} : 2 \][/tex]
where:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (the shorter leg) is [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse (the longest side) is [tex]\(2x\)[/tex].
Given that the hypotenuse is 30, we can set up the equation:
[tex]\[ 2x = 30 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 15 \][/tex]
The longer leg, which is opposite the [tex]\(60^\circ\)[/tex] angle, is:
[tex]\[ x\sqrt{3} = 15\sqrt{3} \][/tex]
Therefore, the length of the longer leg is:
[tex]\[ 15\sqrt{3} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
[tex]\[ 1 : \sqrt{3} : 2 \][/tex]
where:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (the shorter leg) is [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse (the longest side) is [tex]\(2x\)[/tex].
Given that the hypotenuse is 30, we can set up the equation:
[tex]\[ 2x = 30 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 15 \][/tex]
The longer leg, which is opposite the [tex]\(60^\circ\)[/tex] angle, is:
[tex]\[ x\sqrt{3} = 15\sqrt{3} \][/tex]
Therefore, the length of the longer leg is:
[tex]\[ 15\sqrt{3} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
