Question

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. $sqrt{3}:2$ b. $3sqrt{3}:6$ c. $1:sqrt{3}$ d. $sqrt{3}:sqrt{3}$ e. $3:2sqrt{3}$ f. $sqrt{2}:sqrt{3}$

Explanation:

Step1: Recall side - length ratios of 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the shorter leg (opposite the 30 - degree angle) has length $x$, the longer leg (opposite the 60 - degree angle) has length $x\sqrt{3}$, and the hypotenuse has length $2x$. So the ratio of the length of the longer leg to the length of the hypotenuse is $\frac{x\sqrt{3}}{2x}=\frac{\sqrt{3}}{2}$.

Step2: Simplify each option

Option A:

The ratio $\sqrt{3}:2$ is already in the form $\frac{\sqrt{3}}{2}$, so it is correct.

Option B:

Simplify the ratio $3\sqrt{3}:6$. Divide both terms by 3, we get $\frac{3\sqrt{3}}{3}:\frac{6}{3}=\sqrt{3}:2=\frac{\sqrt{3}}{2}$, so it is correct.

Option C:

The ratio $1:\sqrt{3}=\frac{1}{\sqrt{3}}$, which is not equal to $\frac{\sqrt{3}}{2}$, so it is incorrect.

Option D:

The ratio $\sqrt{3}:\sqrt{3} = 1$, which is not equal to $\frac{\sqrt{3}}{2}$, so it is incorrect.

Option E:

Simplify the ratio $3:2\sqrt{3}$. Multiply both the numerator and denominator by $\sqrt{3}$ to get $\frac{3\sqrt{3}}{2\times3}=\frac{\sqrt{3}}{2}$, so it is correct.

Option F:

The ratio $\sqrt{2}:\sqrt{3}=\frac{\sqrt{2}}{\sqrt{3}}$, which is not equal to $\frac{\sqrt{3}}{2}$, so it is incorrect.

Answer:

A. $\sqrt{3}:2$, B. $3\sqrt{3}:6$, E. $3:2\sqrt{3}$