Question

find n.
write your answer in simplest radical form.
meters

Explanation:

Step1: Use the property of 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the side lengths are in the ratio \(1:\sqrt{3}:2\). The side opposite the 30° angle is the shortest side, the side opposite the 60° angle is \(\sqrt{3}\) times the shortest side, and the hypotenuse (opposite the 90° angle) is twice the shortest side. Here, the side of length \(\sqrt{2}\) m is opposite the 60° angle. Let the hypotenuse be \(n\).
We know that \(\cos60^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}\) or \(\sin30^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}\). Also, using the ratio: if the side opposite 60° is \(a = \sqrt{2}\), and the formula for the side opposite 60° in a 30 - 60 - 90 triangle is \(a=\frac{\sqrt{3}}{2}n\) (where \(n\) is the hypotenuse).
Another way: \(\cos60^{\circ}=\frac{\sqrt{2}}{n}\), since \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) and \(\cos60^{\circ}=\frac{1}{2}\).

Step2: Solve for \(n\)

From \(\frac{1}{2}=\frac{\sqrt{2}}{n}\), cross - multiply gives \(n = 2\sqrt{2}\)

Answer:

\(2\sqrt{2}\) meters