Question

find c.
image of a right triangle with angles 60°, 30°, right angle, one leg 9 cm, side c
write your answer in simplest radical form.
blank centimeters
radical symbol 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). The side of length 9 cm is opposite the 60° angle, and \(c\) is opposite the 30° angle? Wait, no—wait, the right angle, 30°, 60°. Let's label: the right angle, 30° at the bottom, 60° at the top. So the side adjacent to 30° (wait, no: the side with length 9 cm is one leg, \(c\) is another leg? Wait, no, in a right triangle, the hypotenuse is opposite the right angle. Wait, no, the right angle is at the left, so the legs are the left leg (9 cm) and the vertical leg (\(c\)), and the hypotenuse is the other side. Wait, no, the angles: 30° at the bottom, 60° at the top, right angle at left. So the side opposite 30° is the left leg (9 cm)? Wait, no, opposite 30° would be the shorter leg. Wait, 30-60-90 triangle: shorter leg (opposite 30°) is \(x\), longer leg (opposite 60°) is \(x\sqrt{3}\), hypotenuse is \(2x\). Wait, in the diagram, the left leg (9 cm) is opposite the 60° angle? Wait, no, angle at top is 60°, so the side opposite 60° is the left leg (9 cm). So longer leg (opposite 60°) is \(x\sqrt{3}\), shorter leg (opposite 30°) is \(x\), hypotenuse \(2x\). So if the longer leg (opposite 60°) is 9 cm, then \(x\sqrt{3} = 9\), so \(x = \frac{9}{\sqrt{3}} = 3\sqrt{3}\)? Wait, no, wait \(c\) is the side opposite 30°, so \(c\) is the shorter leg. Wait, no, let's use trigonometry. Let's take the angle 30°: adjacent side is \(c\), opposite side is 9 cm? Wait, no, the right angle is at left, so the legs are: horizontal leg (wait, no, the left leg is vertical? Wait, the diagram: right angle at left, so the two legs are left (vertical) and bottom (horizontal)? No, the diagram shows a right triangle with right angle at left, 60° at top, 30° at bottom. So the sides: left leg (vertical) is adjacent to 30° angle, bottom leg (horizontal) is opposite 30° angle? Wait, no, angle at bottom is 30°, so the side opposite 30° is the left leg (vertical), and the side opposite 60° is the bottom leg (9 cm). Wait, that makes sense. So in 30-60-90 triangle: opposite 30°: \(x\) (left leg, \(c\)), opposite 60°: \(x\sqrt{3}\) (bottom leg, 9 cm), hypotenuse: \(2x\). So if \(x\sqrt{3} = 9\), then \(x = \frac{9}{\sqrt{3}} = 3\sqrt{3}\)? Wait, no, that can't be. Wait, no, maybe I mixed up. Let's use cosine or sine. Let's take angle 30°: cos(30°) = adjacent / hypotenuse, but maybe better to use sine. Sine of 30° is opposite / hypotenuse, but opposite to 30° is the left leg (\(c\)), adjacent is the bottom leg (9 cm). Wait, no, angle at bottom is 30°, so the sides: opposite 30° is left leg (\(c\)), adjacent is bottom leg (9 cm)? No, adjacent to 30° is the side next to it, which is the bottom leg (9 cm), and opposite is left leg (\(c\)). So tan(30°) = opposite / adjacent = \(c / 9\). Tan(30°) is \(1/\sqrt{3}\), so \(c = 9 \times \tan(30°) = 9 \times \frac{1}{\sqrt{3}} = 3\sqrt{3}\)? Wait, no, that's not right. Wait, maybe I got the angles wrong. Wait, angle at top is 60°, so the side opposite 60° is the bottom leg (9 cm), and side opposite 30° is \(c\). So in 30-60-90 triangle, the side opposite 60° is \(x\sqrt{3}\), side opposite 30° is \(x\), hypotenuse \(2x\). So if \(x\sqrt{3} = 9\), then \(x = 9 / \sqrt{3} = 3\sqrt{3}\). Wait, but that would make \(c = 3\sqrt{3}\), but that seems short. Wait, no, maybe the 9 cm is the hypotenuse? Wait, no, the right angle is at left, so hypotenuse is the side opposite right angle, which is the top side (connecting 60° and 30°). Wait, I think I made a mistake. Let's re-express:

In a 30-60-90 right triangle, the ratios are:

- Shorter leg (opposite 30°): \(x\)

- Longer leg (opposite 60°): \(x\sqrt{3}\)

- Hypotenuse: \(2x\)

Looking at the diagram: the right angle is at the left, 60° at the top, 30° at the bottom. So the side opposite 30° is the left leg (vertical), and the side opposite 60° is the bottom leg (horizontal, 9 cm). So the bottom leg (9 cm) is the longer leg (opposite 60°), so \(x\sqrt{3} = 9\), so \(x = 9 / \sqrt{3} = 3\sqrt{3}\). Wait, but that would mean the shorter leg (opposite 30°) is \(3\sqrt{3}\), but that's \(c\)? Wait, no, \(c\) is the vertical leg, which is opposite 30°, so yes. Wait, but maybe the 9 cm is the hypotenuse? Wait, no, the hypotenuse is opposite the right angle, so the hypotenuse is the side connecting the 60° and 30° angles. Let's check with trigonometry. Let's take the angle 60°: sine(60°) = opposite / hypotenuse. Opposite to 60° is the bottom leg (9 cm), so sine(60°) = 9 / hypotenuse. Hypotenuse = 9 / (√3/2) = 18/√3 = 6√3. Then, the side \(c\) (opposite 30°) is hypotenuse * sine(30°) = 6√3 * 1/2 = 3√3. Wait, that's the same as before. Wait, but maybe I messed up the labels. Alternatively, using cosine for angle 30°: cos(30°) = adjacent / hypotenuse. Adjacent to 30° is \(c\), hypotenuse is 6√3, so cos(30°) = \(c\) / 6√3. \(c = 6√3 * (√3/2) = 6*3/2 = 9\)? Wait, no, that's conflicting. Wait, no, I think I mixed up adjacent and opposite. Let's clarify the triangle:

- Right angle at (0,0)

- 30° angle at (0, c)

- 60° angle at (9, c)

Wait, no, the diagram shows the right angle at the left, 60° at the top, 30° at the bottom. So the vertices are: left (right angle), top (60°), bottom (30°). So the sides:

- Left to top: vertical leg (\(c\))

- Left to bottom: horizontal leg (9 cm)

- Top to bottom: hypotenuse

So angle at bottom (30°): the sides adjacent to 30° are the horizontal leg (9 cm) and the hypotenuse? No, adjacent is the horizontal leg (9 cm), opposite is the vertical leg (\(c\)). So tan(30°) = opposite / adjacent = \(c\) / 9. So \(c = 9 * tan(30°) = 9 * (1/√3) = 3√3\). Ah, that's correct. So \(c = 3\sqrt{3}\)? Wait, no, tan(30°) is 1/√3, so 9*(1/√3) = 3√3. Yes. So that's the answer.

Step2: Simplify

We had \(c = 9 \times \frac{1}{\sqrt{3}} = \frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}\). Wait, no, wait, if we use tan(30°) = opposite/adjacent, and opposite is \(c\), adjacent is 9 cm, then \(tan(30°) = c / 9\), so \(c = 9 \times tan(30°) = 9 \times \frac{1}{\sqrt{3}} = 3\sqrt{3}\). Alternatively, using the 30-60-90 ratios: the side opposite 60° is 9 cm (longer leg), so longer leg = \(x\sqrt{3}\), so \(x\sqrt{3} = 9\), so \(x = 3\sqrt{3}\) (shorter leg, opposite 30°), which is \(c\).

Answer:

\(3\sqrt{3}\)