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In a [tex]30^{\circ}-60^{\circ}-90^{\circ}[/tex] triangle, the length of the hypotenuse is 30. Find the length of the longer leg.
A. 15 B. [tex]10 \sqrt{3}[/tex] C. [tex]15 \sqrt{2}[/tex] D. [tex]15 \sqrt{3}[/tex]
Please select the best answer from the choices provided: A B C D
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:
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