In a 30-60-90 triangle, which side length corresponds to the angle opposite 60 degrees?

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Multiple Choice

In a 30-60-90 triangle, which side length corresponds to the angle opposite 60 degrees?

In a 30-60-90 triangle, the relationship between the lengths of the sides is defined by specific ratios. The triangle has side lengths proportional to 1 (opposite the 30-degree angle), √3 (opposite the 60-degree angle), and 2 (the hypotenuse). When we analyze what each side represents based on the given angle measures: - The side opposite the 30-degree angle is the shortest and corresponds to the length of \( n \). - The hypotenuse, which is opposite the right angle, is the longest side and measures \( 2n \). - The side opposite the 60-degree angle is thus the middle length in this ratio, which corresponds to \( n\sqrt{3} \). Therefore, if \( n \) represents the length of the side opposite the 30-degree angle, the side opposite the 60-degree angle will indeed be \( n\sqrt{3} \). This makes option C the correct choice. The presence of the radical indicates the relationship captured by the properties of a 30-60-90 triangle, where the longer leg (opposite the 60-degree angle) is \(\sqrt{3}\) times the shorter leg (

In a 30-60-90 triangle, the relationship between the lengths of the sides is defined by specific ratios. The triangle has side lengths proportional to 1 (opposite the 30-degree angle), √3 (opposite the 60-degree angle), and 2 (the hypotenuse).

When we analyze what each side represents based on the given angle measures:

  • The side opposite the 30-degree angle is the shortest and corresponds to the length of ( n ).

  • The hypotenuse, which is opposite the right angle, is the longest side and measures ( 2n ).

  • The side opposite the 60-degree angle is thus the middle length in this ratio, which corresponds to ( n\sqrt{3} ).

Therefore, if ( n ) represents the length of the side opposite the 30-degree angle, the side opposite the 60-degree angle will indeed be ( n\sqrt{3} ). This makes option C the correct choice. The presence of the radical indicates the relationship captured by the properties of a 30-60-90 triangle, where the longer leg (opposite the 60-degree angle) is (\sqrt{3}) times the shorter leg (

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