Question

find t.
image of a right triangle with a right angle, one angle 60°, one angle 30°, the side opposite 60° is 13√3 mi, and the side adjacent to 60° (and opposite 30°) is t
write your answer in simplest radical form.
blank box miles
√ button

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 \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Step2: Determine which side is given

The side given is \(13\sqrt{3}\) miles, and it is opposite the \(60^{\circ}\) angle? Wait, no. Wait, the right angle, the \(30^{\circ}\) angle, and the \(60^{\circ}\) angle. Let's label the triangle: let the right angle be at the left, the \(30^{\circ}\) angle at the bottom, \(60^{\circ}\) at the top. So the side adjacent to \(30^{\circ}\) and opposite \(60^{\circ}\) is \(13\sqrt{3}\), and the side \(t\) is opposite the \(30^{\circ}\) angle? Wait, no, wait. Wait, in a 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is the shortest side (let's say length \(t\)), the side opposite \(60^{\circ}\) is \(t\sqrt{3}\), and hypotenuse is \(2t\). Wait, the side given is \(13\sqrt{3}\), which is opposite the \(60^{\circ}\) angle? Wait, no, the side labeled \(13\sqrt{3}\) is adjacent to the \(30^{\circ}\) angle? Wait, maybe I got the labels wrong. Let's re - label: the right angle is at the left, so the two legs are: one leg (let's say vertical) opposite \(60^{\circ}\), and the other leg (horizontal, \(t\)) opposite \(30^{\circ}\), and the side \(13\sqrt{3}\) is the leg opposite \(60^{\circ}\). Wait, in a 30 - 60 - 90 triangle, \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{t}{13\sqrt{3}}\). Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), so \(\frac{t}{13\sqrt{3}}=\frac{1}{\sqrt{3}}\). Multiply both sides by \(13\sqrt{3}\): \(t = 13\). Wait, or maybe using the ratio. Let's think again. The side opposite \(30^{\circ}\) is \(t\), the side opposite \(60^{\circ}\) is \(t\sqrt{3}\). The side given is \(13\sqrt{3}\), which is equal to \(t\sqrt{3}\)? Wait, no, if the side opposite \(60^{\circ}\) is \(t\sqrt{3}\), and that side is \(13\sqrt{3}\), then \(t\sqrt{3}=13\sqrt{3}\), so \(t = 13\)? Wait, no, maybe the side \(13\sqrt{3}\) is the leg adjacent to \(60^{\circ}\). Wait, let's use trigonometry. \(\tan(60^{\circ})=\frac{13\sqrt{3}}{t}\), since \(\tan(60^{\circ})=\sqrt{3}\), so \(\sqrt{3}=\frac{13\sqrt{3}}{t}\). Cross - multiply: \(t\sqrt{3}=13\sqrt{3}\), then divide both sides by \(\sqrt{3}\), we get \(t = 13\). Alternatively, using the 30 - 60 - 90 ratio: the side opposite \(30^{\circ}\) is \(t\), side opposite \(60^{\circ}\) is \(t\sqrt{3}\). The side given is \(13\sqrt{3}\), which is equal to \(t\sqrt{3}\), so \(t = 13\).

Answer:

\(13\)