Question

question 8 of 10
which of the following could be the ratio of the length of the longer leg of a 30 - 60 - 90 triangle to the length of its hypotenuse?
check all that apply.
a. $sqrt{2}:sqrt{3}$
b. $sqrt{3}:sqrt{3}$
c. $sqrt{3}:2$
d. $2:2sqrt{2}$
e. $1:sqrt{3}$
f. $2sqrt{3}:4$

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is the shortest leg (\(1\)), the side opposite 60° is the longer leg (\(\sqrt{3}\)), and the hypotenuse is \(2\). So the ratio of the longer leg to the hypotenuse is \(\sqrt{3}:2\).

Step2: Analyze each option

• Option A: \(\sqrt{2}:\sqrt{3}\) simplifies to \(\frac{\sqrt{2}}{\sqrt{3}}\), not equal to \(\frac{\sqrt{3}}{2}\).
• Option B: \(\sqrt{3}:\sqrt{3}=1:1\), not equal to \(\frac{\sqrt{3}}{2}\).
• Option C: \(\sqrt{3}:2\) is exactly the ratio of longer leg to hypotenuse, so this is correct.
• Option D: \(2:2\sqrt{2}=\frac{2}{2\sqrt{2}}=\frac{1}{\sqrt{2}}\), not equal to \(\frac{\sqrt{3}}{2}\).
• Option E: \(1:\sqrt{3}=\frac{1}{\sqrt{3}}\), not equal to \(\frac{\sqrt{3}}{2}\).
• Option F: \(2\sqrt{3}:4=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\), which is the same as the ratio of longer leg to hypotenuse, so this is correct. Also, simplify \(\sqrt{3}:2\) and \(2\sqrt{3}:4\) (divide numerator and denominator of \(2\sqrt{3}:4\) by 2, we get \(\sqrt{3}:2\)), and check if the ratios match the 30-60-90 triangle ratio.

Answer:

C. \(\sqrt{3} : 2\), F. \(2\sqrt{3} : 4\)