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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


Sagot :

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]