Question

find the length of side x in simplest radical form with a rational denominator. answer attempt 2 out of 2 x =

Explanation:

Step1: Use cosine function

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 60^{\circ}$, the adjacent side to the $60^{\circ}$ angle is 9 and the hypotenuse is $x$. So, $\cos60^{\circ}=\frac{9}{x}$.
Since $\cos60^{\circ}=\frac{1}{2}$, we have $\frac{1}{2}=\frac{9}{x}$.

Step2: Solve for $x$

Cross - multiply to get $x = 18$.
We can also use the relationship for a 30 - 60 - 90 triangle. In a 30 - 60 - 90 triangle, if the side opposite the $30^{\circ}$ angle is $a$, the side opposite the $60^{\circ}$ angle is $a\sqrt{3}$ and the hypotenuse is $2a$. Here, the side opposite the $60^{\circ}$ angle is 9. Let the side opposite the $30^{\circ}$ angle be $y$. Then $y\sqrt{3}=9$, so $y = 3\sqrt{3}$, and the hypotenuse $x = 2y=6\sqrt{3}$.
We'll use the trigonometric approach.

Answer:

$6\sqrt{3}$