Question

find d.
right triangle with right angle, one leg labeled 20√3 km, angles 30°, 60°, and hypotenuse labeled d
write your answer in simplest radical form.
blank kilometers
√ button

Explanation:

Step1: Identify triangle type

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

Step2: Relate given side to ratio

The side opposite 30° is \(20\sqrt{3}\) km? Wait, no—wait, the side adjacent to 30° (or opposite 60°) is \(20\sqrt{3}\) km. Wait, let's label: the right angle, 30°, 60°. The side opposite 30° is the shortest leg, opposite 60° is longer leg, hypotenuse is \(d\). Wait, the leg given is \(20\sqrt{3}\) km, which is opposite 30°? No, wait, angle 30°: the side adjacent to 30° is the leg with \(20\sqrt{3}\)? Wait, no, let's use trigonometry. Let's take angle 30°: \(\cos(30°)=\frac{\text{adjacent}}{\text{hypotenuse}}\), but adjacent is \(20\sqrt{3}\), hypotenuse is \(d\)? Wait, no, wait the right angle is at the top, so the legs are: one leg is \(20\sqrt{3}\) (let's say adjacent to 30°), and the other leg is opposite 30°. Wait, \(\sin(30°)=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos(30°)=\frac{\text{adjacent}}{\text{hypotenuse}}\). Wait, angle 30°: adjacent side is \(20\sqrt{3}\), hypotenuse is \(d\). So \(\cos(30°)=\frac{20\sqrt{3}}{d}\). But \(\cos(30°)=\frac{\sqrt{3}}{2}\). So \(\frac{\sqrt{3}}{2}=\frac{20\sqrt{3}}{d}\). Cross-multiplying: \(d\times\sqrt{3}=2\times20\sqrt{3}\). Divide both sides by \(\sqrt{3}\): \(d = 40\). Wait, or maybe using 30-60-90 ratios: in 30-60-90, the sides are \(x\), \(x\sqrt{3}\), \(2x\), where \(x\) is opposite 30°, \(x\sqrt{3}\) opposite 60°, \(2x\) hypotenuse. Wait, the leg given is \(20\sqrt{3}\), which is \(x\sqrt{3}\) (opposite 60°), so \(x\sqrt{3}=20\sqrt{3}\) ⇒ \(x = 20\). Then hypotenuse \(d = 2x = 40\). Yes, that matches. So \(d = 40\) km.

Step3: Verify

Using trigonometry: \(\sin(60°)=\frac{20\sqrt{3}}{d}\), \(\sin(60°)=\frac{\sqrt{3}}{2}\), so \(\frac{\sqrt{3}}{2}=\frac{20\sqrt{3}}{d}\) ⇒ \(d = 40\). Correct.

Answer:

\(40\)