Question

find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle. In such a triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite \(30^\circ\) is the shortest, opposite \(60^\circ\) is \(\sqrt{3}\) times the shortest, and the hypotenuse is twice the shortest. Here, the side of length 9 is opposite the \(30^\circ\) angle? Wait, no—wait, the right angle, \(60^\circ\), and \(30^\circ\). Wait, the side with length 9: let's check angles. The angle of \(60^\circ\): the side opposite \(60^\circ\) would be... Wait, no, let's label the triangle. Let's denote the right angle as \(C\), \(60^\circ\) as \(A\), \(30^\circ\) as \(B\). Then side opposite \(A\) (60°) is \(BC\), opposite \(B\) (30°) is \(AC\), and hypotenuse \(AB = x\). Wait, the given side is 9, which is \(AC\) (adjacent to 60°? Wait, no, the side labeled 9 is one leg, with the right angle. Wait, the triangle has a right angle, 60°, 30°, so it's a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in the ratio: shorter leg (opposite 30°) : longer leg (opposite 60°) : hypotenuse = \(1 : \sqrt{3} : 2\).

Wait, the side with length 9: let's see, which angle is it opposite to? The angle of 60°: the side opposite 60° is the longer leg, and the side opposite 30° is the shorter leg. Wait, the right angle is between the two legs. So if one leg is 9, and the angles are 60° and 30°, then the leg opposite 30° is the shorter leg, and the leg opposite 60° is the longer leg. Wait, let's clarify: in a 30-60-90 triangle, the side opposite 30° is the shortest (let's call it \(a\)), the side opposite 60° is \(a\sqrt{3}\), and the hypotenuse is \(2a\).

Looking at the triangle, the side labeled 9: let's see which angle it's adjacent to. The angle of 60°: the side adjacent to 60° would be the shorter leg (opposite 30°), and the side opposite 60° would be the longer leg. Wait, no—wait, the right angle is at the top, so the two legs are the vertical and horizontal sides, and the hypotenuse is \(x\). The angle at the bottom right is 30°, so the side opposite 30° is the vertical leg (length 9), and the side opposite 60° is the horizontal leg, and the hypotenuse is \(x\).

Ah, that makes sense. So the side opposite 30° is 9, so that's the shorter leg (\(a = 9\)). Then the hypotenuse \(x\) is \(2a\)? Wait, no—wait, no: in a 30-60-90 triangle, the hypotenuse is twice the shorter leg (opposite 30°). Wait, if the shorter leg (opposite 30°) is \(a\), then hypotenuse is \(2a\), and the longer leg (opposite 60°) is \(a\sqrt{3}\).

Wait, in the triangle, the angle at the bottom right is 30°, so the side opposite 30° is the leg with length 9? Wait, no, the leg with length 9 is adjacent to the 60° angle. Wait, maybe I got the angles reversed. Let's re-express: the triangle has angles 90°, 60°, 30°. So the side opposite 30° is the shortest leg, side opposite 60° is the longer leg, hypotenuse is the longest.

Looking at the triangle, the side labeled 9: let's see, the angle at the bottom left is 60°, so the side opposite 60° is the horizontal leg (opposite 60°), and the side opposite 30° is the vertical leg (length 9). Wait, no—wait, the right angle is at the top, so the two legs are: one leg is 9 (vertical), the other leg is the horizontal leg (let's call it \(b\)), and hypotenuse \(x\). The angle at the bottom right is 30°, so the angle at the bottom right (30°) has its opposite side as the vertical leg (length 9). So the side opposite 30° is 9, which is the shorter leg. Therefore, the hypotenuse \(x\) is twice the shorter leg? Wait, no—wait, no: the shorter leg is opposite 30°, so if the shorter leg is \(a\), then hypotenuse is \(2a\). But if the shorter leg is 9, then hypotenuse would be \(18\)? Wait, that can't be, because the hypotenuse should be longer than either leg. Wait, but 9 is a leg, and \(x\) is the hypotenuse, so \(x\) should be longer than 9. Wait, but maybe I mixed up the legs.

Wait, let's use trigonometry. In the right triangle, we can use sine or cosine. Let's take the angle of 30°: \(\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{x}\). Since \(\sin(30^\circ) = \frac{1}{2}\), so \(\frac{1}{2} = \frac{9}{x}\), which gives \(x = 18\)? Wait, but that seems too simple. Wait, no—wait, maybe the side labeled 9 is the longer leg (opposite 60°). Let's check that. If the side opposite 60° is 9, then \(\sin(60^\circ) = \frac{9}{x}\). Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), then \(\frac{\sqrt{3}}{2} = \frac{9}{x}\), so \(x = \frac{18}{\sqrt{3}} = 6\sqrt{3}\)? Wait, that contradicts the previous. So which is it?

Wait, let's look at the triangle again. The right angle is at the top, so the two legs are the left leg (length 9) and the bottom leg (horizontal), and the hypotenuse is \(x\) (the base). The angle at the bottom left is 60°, so the angle at the bottom left (60°) has the left leg (length 9) as its adjacent side, and the bottom leg as its opposite side. The angle at the bottom right is 30°, so the left leg (length 9) is its opposite side.

So for the angle of 30°: \(\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{x}\). Since \(\sin(30^\circ) = \frac{1}{2}\), then:

\(\frac{1}{2} = \frac{9}{x}\)

Solving for \(x\):

\(x = 9 \times 2 = 18\)

Wait, but that would mean the hypotenuse is 18, and the other leg (opposite 60°) would be \(18 \times \sin(60^\circ) = 18 \times \frac{\sqrt{3}}{2} = 9\sqrt{3}\). But in the triangle, the left leg is 9, which is opposite 30°, so that's correct. Wait, but let's confirm with the 30-60-90 ratios. If the shorter leg (opposite 30°) is 9, then the hypotenuse is \(2 \times 9 = 18\), and the longer leg (opposite 60°) is \(9\sqrt{3}\). That matches. So the hypotenuse \(x\) is 18? Wait, but that seems too straightforward. Wait, maybe I made a mistake in identifying the opposite side. Let's check the angle of 60°: \(\sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{bottom leg}}{x}\). The bottom leg would be \(x \times \sin(60^\circ)\), and the left leg (9) is \(x \times \sin(30^\circ)\). Since \(\sin(30^\circ) = \frac{1}{2}\), then \(9 = x \times \frac{1}{2}\), so \(x = 18\). Yes, that's correct. So the length of \(x\) is 18.

Wait, but let's re-express. In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is the shortest (1 part), opposite 60° is \(\sqrt{3}\) parts, and hypotenuse is 2 parts. So if the side opposite 30° is 9, then 1 part = 9, so hypotenuse (2 parts) is \(2 \times 9 = 18\). That's correct. So the hypotenuse \(x\) is 18.

Step1: Identify the triangle type and ratios

This is a 30-60-90 right triangle. The side ratios are: shorter leg (opposite 30°) : longer leg (opposite 60°) : hypotenuse = \(1 : \sqrt{3} : 2\).

Step2: Determine the side opposite 30°

The side labeled 9 is opposite the 30° angle (shorter leg). Let the shorter leg length be \(a = 9\).

Step3: Calculate the hypotenuse \(x\)

In a 30-60-90 triangle, the hypotenuse is twice the shorter leg:
\(x = 2a\)
Substitute \(a = 9\):
\(x = 2 \times 9 = 18\)

Answer:

\(18\)