Question

question 3 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. ( 1:sqrt{3} )
b. ( sqrt{2}:sqrt{2} )
c. ( sqrt{3}:3 )
d. ( sqrt{3}:sqrt{3} )
e. ( sqrt{2}:sqrt{3} )
f. ( 1:sqrt{2} )

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\) (shorter leg : longer leg : hypotenuse). The two legs (the non - hypotenuse sides) have lengths in the ratio \(1:\sqrt{3}\) or \(\sqrt{3}:3\) (since \(\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\) after rationalizing or simplifying the ratio).

Step2: Analyze each option

• Option A: The ratio is \(1:\sqrt{3}\), which matches the ratio of the legs of a 30 - 60 - 90 triangle. So this option is correct.
• Option B: The ratio \(\sqrt{2}:\sqrt{2} = 1:1\), which is the ratio of the legs of an isosceles right triangle (45 - 45 - 90 triangle), not a 30 - 60 - 90 triangle. So this option is incorrect.
• Option C: The ratio \(\sqrt{3}:3=\frac{\sqrt{3}}{3}:1 = 1:\sqrt{3}\) (dividing both parts by \(\sqrt{3}\)), which is equivalent to the ratio of the legs of a 30 - 60 - 90 triangle. So this option is correct.
• Option D: The ratio \(\sqrt{3}:\sqrt{3}=1:1\), which is for a 45 - 45 - 90 triangle, not a 30 - 60 - 90 triangle. So this option is incorrect.
• Option E: The ratio \(\sqrt{2}:\sqrt{3}\) does not match the \(1:\sqrt{3}\) ratio of the legs of a 30 - 60 - 90 triangle. So this option is incorrect.
• Option F: The ratio \(1:\sqrt{2}\) is for a 45 - 45 - 90 triangle (since in a 45 - 45 - 90 triangle, legs are equal and hypotenuse is \(\sqrt{2}\) times the leg, so leg ratio is \(1:1\) and leg to hypotenuse is \(1:\sqrt{2}\)), not a 30 - 60 - 90 triangle. So this option is incorrect.

Answer:

A. \(1:\sqrt{3}\), C. \(\sqrt{3}:3\)