Question

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

Explanation:

Step1: Recall the sine function

In a right - triangle, the sine of an angle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the opposite side to the \(30^{\circ}\) angle is \(x\), and the hypotenuse is \(5\).
So, \(\sin30^{\circ}=\frac{x}{5}\).

Step2: Substitute the value of \(\sin30^{\circ}\)

Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{x}{5}\).
Cross - multiply to get \(x=\frac{5}{2}\).

Another way using the \(30 - 60-90\) triangle ratio:
In a \(30 - 60 - 90\) triangle, if the hypotenuse \(c\) and the side opposite the \(30^{\circ}\) angle \(a\), the ratio is \(a:b:c = 1:\sqrt{3}:2\) (where \(a\) is opposite \(30^{\circ}\), \(b\) is opposite \(60^{\circ}\), \(c\) is the hypotenuse).
Here \(c = 5\), and \(x\) is opposite \(30^{\circ}\). Let \(x=a\), then \(a=\frac{c}{2}\) (because \(c = 2a\) from the ratio \(a:c=1:2\)).

Answer:

\(\frac{5}{2}\)