Question

1. solve for x image of a right triangle with one leg labeled 21, one angle labeled 30°, and the other leg labeled x

Explanation:

Step1: Identify triangle type

This is a right - triangle with one angle \(30^{\circ}\), so it's a 30 - 60 - 90 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 \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\). Here, the side of length 21 is adjacent to the \(30^{\circ}\) angle? Wait, no. Wait, the right angle, the \(30^{\circ}\) angle, so the side opposite \(30^{\circ}\) is \(x\), and the side adjacent to \(30^{\circ}\) (the leg) is 21? Wait, no, in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 30^{\circ}\), opposite side is \(x\), adjacent side is 21. So \(\tan(30^{\circ})=\frac{x}{21}\). Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\), we can solve for \(x\).

Step2: Solve for \(x\)

We have the equation \(\tan(30^{\circ})=\frac{x}{21}\). Substitute \(\tan(30^{\circ})=\frac{\sqrt{3}}{3}\) into the equation:
\(\frac{\sqrt{3}}{3}=\frac{x}{21}\)
Cross - multiply: \(x = 21\times\frac{\sqrt{3}}{3}\)
Simplify: \(x = 7\sqrt{3}\approx12.12\) (or we can also use the 30 - 60 - 90 triangle ratio. In a 30 - 60 - 90 triangle, if the side opposite \(60^{\circ}\) is 21, then the side opposite \(30^{\circ}\) (which is \(x\)) is \(\frac{21}{\sqrt{3}}=\frac{21\sqrt{3}}{3}=7\sqrt{3}\), because in 30 - 60 - 90 triangle, the side opposite \(60^{\circ}\) is \(a\sqrt{3}\) and the side opposite \(30^{\circ}\) is \(a\). So if \(a\sqrt{3}=21\), then \(a=\frac{21}{\sqrt{3}} = 7\sqrt{3}\), which is \(x\)).

Answer:

\(x = 7\sqrt{3}\) (or approximately \(12.12\))