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The longer leg of a right triangle has a length of 15. One angle in the triangle is 60 degrees. What is the length of the shortest leg?

A. [tex]5 \sqrt{3}[/tex]
B. [tex]10 \sqrt{3}[/tex]
C. [tex]15 \sqrt{3}[/tex]
D. 5


Sagot :

To determine the length of the shortest leg in the given right triangle, we need to utilize properties of a 30-60-90 triangle.

A 30-60-90 triangle has side lengths that are consistent based on the following ratios:
- The length of the shorter leg is [tex]\(a\)[/tex]
- The length of the longer leg (adjacent to the 60-degree angle) is [tex]\(a \sqrt{3}\)[/tex]
- The hypotenuse is [tex]\(2a\)[/tex]

Given that the length of the longer leg is 15, we can set up an equation based on the known ratios:

[tex]\[ \text{longer leg} = a \sqrt{3} \][/tex]

[tex]\[ 15 = a \sqrt{3} \][/tex]

To solve for [tex]\(a\)[/tex], we divide both sides of the equation by [tex]\(\sqrt{3}\)[/tex]:

[tex]\[ a = \frac{15}{\sqrt{3}} \][/tex]

Rationalizing the denominator, we multiply the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:

[tex]\[ a = \frac{15 \sqrt{3}}{3} \][/tex]

[tex]\[ a = 5 \sqrt{3} \][/tex]

Thus, the length of the shortest leg is [tex]\(5 \sqrt{3}\)[/tex].

Hence, the correct answer is:
[tex]\[ \boxed{5 \sqrt{3}} \][/tex]