Question

question 2 of 10

which of the following could be the ratio between the lengths of the two legs of a 30-60-90 triangle?

check all that apply.

a. \\(\sqrt{2} : \sqrt{2}\\)
b. \\(1 : \sqrt{3}\\)
c. \\(\sqrt{2} : \sqrt{3}\\)
d. \\(\sqrt{3} : \sqrt{3}\\)
e. \\(2\sqrt{3} : 6\\)
f. \\(1 : \sqrt{2}\\)

Explanation:

Identify the side length ratios of a 30-60-90 triangle

Using the Special Right Triangles knowledge point

$$ \text{Ratio of sides: } 1 : \sqrt{3} : 2 $$

$$ \text{Shorter leg: } x, \quad \text{Longer leg: } x\sqrt{3}, \quad \text{Hypotenuse: } 2x $$

Determine the ratio between the lengths of the two legs

Using the Special Right Triangles knowledge point

$$ \text{Ratio of legs: } \frac{\text{shorter leg}}{\text{longer leg}} = \frac{1}{\sqrt{3}} \quad \text{or} \quad \frac{\text{longer leg}}{\text{shorter leg}} = \frac{\sqrt{3}}{1} $$

$$ \text{Target ratios: } 1 : \sqrt{3} \quad \text{or} \quad \sqrt{3} : 1 $$

Evaluate each given option

Using the Special Right Triangles knowledge point

$$
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Answer:

• A. \(\sqrt{2} : \sqrt{2}\)
• B. \(1 : \sqrt{3}\) (Correct answer)
• C. \(\sqrt{2} : \sqrt{3}\)
• D. \(\sqrt{3} : \sqrt{3}\)
• E. \(2\sqrt{3} : 6\) (Correct answer)
• F. \(1 : \sqrt{2}\)