Question

find n.
right triangle with right angle at top left, horizontal leg labeled n, vertical leg labeled 10√3 km, angle at top right is 60°, angle at bottom is 30°
write your answer in simplest radical form.
blank kilometers

Explanation:

Step1: Identify triangle type

This is a 30 - 60 - 90 right triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite 30°, 60°, 90° respectively). The side opposite 60° is \(10\sqrt{3}\) km, and \(n\) is opposite 30°.

Step2: Use tangent or ratio

We can use the tangent function. \(\tan(30^{\circ})=\frac{n}{10\sqrt{3}}\). Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), we have \(\frac{1}{\sqrt{3}}=\frac{n}{10\sqrt{3}}\). Cross - multiplying gives \(n = \frac{10\sqrt{3}}{\sqrt{3}}=10\). Or using the ratio: in 30 - 60 - 90, side opposite 30° is \(x\), opposite 60° is \(x\sqrt{3}\). Here \(x\sqrt{3}=10\sqrt{3}\), so \(x = 10\).

Answer:

\(10\)