IDNLearn.com makes it easy to get reliable answers from knowledgeable individuals. Whether it's a simple query or a complex problem, our experts have the answers you need.

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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.