Question

find a.
right triangle with angles 30°, 60°, 90°; side adjacent to 30° is $18\sqrt{3}$ cm, side a is opposite to 30°
write your answer in simplest radical form.
blank centimeters
radical symbol button

Explanation:

Step1: Identify trigonometric ratio

In a right - triangle, we can use the tangent function. For the \(30^{\circ}\) angle, the opposite side to \(30^{\circ}\) is \(a\) and the adjacent side is \(18\sqrt{3}\) cm. We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\theta = 30^{\circ}\), \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), and \(\tan(30^{\circ})=\frac{a}{18\sqrt{3}}\).
Alternatively, we can use the fact that in a \(30 - 60-90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite to \(30^{\circ}\), opposite to \(60^{\circ}\), hypotenuse respectively). The side adjacent to \(30^{\circ}\) (which is the side with length \(18\sqrt{3}\)) is opposite to \(60^{\circ}\), and the side \(a\) is opposite to \(30^{\circ}\).
If we let the side opposite to \(30^{\circ}\) be \(x\), the side opposite to \(60^{\circ}\) be \(x\sqrt{3}\), and the hypotenuse be \(2x\). Here, the side opposite to \(60^{\circ}\) is \(18\sqrt{3}\), so \(x\sqrt{3}=18\sqrt{3}\).

Step2: Solve for \(a\) (which is \(x\))

We have the equation \(x\sqrt{3}=18\sqrt{3}\). Divide both sides of the equation by \(\sqrt{3}\):
\(x=\frac{18\sqrt{3}}{\sqrt{3}}\)
The \(\sqrt{3}\) in the numerator and denominator cancels out, so \(x = 18\). Since \(a=x\) (because \(a\) is the side opposite to \(30^{\circ}\)), we get \(a = 18\).

Answer:

\(18\)