Question

question 24 of 25 in the triangle below, what is the length of the side opposite the 30° angle? a. 3√3 b. 3/2 c. √3 d. 2√3

Explanation:

Step1: Recall trigonometric ratio

In a right - triangle, if the angle is $\theta = 30^{\circ}$, and we know the length of the hypotenuse $c$ and want to find the length of the side opposite the angle $a$, we use the sine function $\sin\theta=\frac{a}{c}$. Here, assume the hypotenuse is $c$ and the side opposite the $30^{\circ}$ angle is $a$. Also, if we consider the relationship between the sides of a 30 - 60 - 90 triangle, if the side opposite the $30^{\circ}$ angle is $x$, the side opposite the $60^{\circ}$ angle is $\sqrt{3}x$. Given the side opposite the $60^{\circ}$ angle is 3.

Step2: Set up equation

Let the side opposite the $30^{\circ}$ angle be $x$. Since the side opposite the $60^{\circ}$ angle is $\sqrt{3}x$ and it is equal to 3, we have the equation $\sqrt{3}x = 3$.

Step3: Solve for $x$

Divide both sides of the equation $\sqrt{3}x=3$ by $\sqrt{3}$: $x=\frac{3}{\sqrt{3}}$. Rationalize the denominator: $x = \frac{3\sqrt{3}}{3}=\sqrt{3}$.

Answer:

C. $\sqrt{3}$