Question

find the length of side ( x ) in simplest radical form with a rational denominator.

answer attempt 1 out of 2

( x = )

Explanation:

Step1: Use cosine function

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 60^{\circ}\), adjacent side is \(9\), and 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: \(x = 18\).

Another way:

Step1: Use the relationship of sides in 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 \(2a\).
The side opposite the \(30^{\circ}\) angle: Let the side opposite \(30^{\circ}\) be \(y\). Using \(\sin30^{\circ}=\frac{y}{x}\) and \(\sin30^{\circ}=\frac{1}{2}\), also \(\sin60^{\circ}=\frac{9}{x}\) (\(\sin60^{\circ}=\frac{\sqrt{3}}{2}\)).
Since \(\sin60^{\circ}=\frac{9}{x}\), we have \(x=\frac{9}{\sin60^{\circ}}\).

Step2: Rationalize the denominator

\(x=\frac{9}{\frac{\sqrt{3}}{2}}=\frac{9\times2}{\sqrt{3}}=\frac{18}{\sqrt{3}}\). Multiply numerator and denominator by \(\sqrt{3}\): \(x = 6\sqrt{3}\times\sqrt{3}=18\).

Answer:

\(x = 18\)