Question

1. use the 30° - 60° - 90° triangle pattern to determine the lengths of the missing sides in triangles below. do not use a trig tool or a calculator. leave your answers in exact form.

x = __, y = __
image of a right triangle with one leg 6, angle 60° at the bottom, other leg x, hypotenuse y

Explanation:

Step1: Identify triangle type

This is a 30°-60°-90° triangle. The side opposite 30° is the shortest. Here, the side adjacent to 60° (length 6) is opposite 30°, wait no—wait, the right angle, 60°, so the angles are 90°, 60°, so the third angle is 30°. So the side opposite 30° is 6? Wait no, the side with length 6: let's label the triangle. Right angle, one angle 60°, so the other is 30°. The side adjacent to 60° (the leg) is length 6? Wait, no: in 30-60-90, sides are in ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is the shortest (let's call it \(a\)), opposite 60° is \(a\sqrt{3}\), hypotenuse is \(2a\).

Looking at the triangle: the leg with length 6 is adjacent to 60°, so it's opposite 30°? Wait, no. Let's see: the right angle, angle 60°, so the angle opposite the side of length 6 is 30°? Wait, the side labeled 6: let's check the angles. The right angle, angle 60°, so the third angle is 30°. So the side opposite 30° is 6? Wait, no, the side with length 6: if the angle is 60°, then the side adjacent to 60° (the leg) is 6? Wait, no, maybe I got the sides mixed. Let's re-express:

In a 30-60-90 triangle, the sides are:
- Short leg (opposite 30°): \(a\)
- Long leg (opposite 60°): \(a\sqrt{3}\)
- Hypotenuse: \(2a\)

Looking at the triangle, the leg with length 6: let's see which angle it's opposite. The angle of 60°: the side opposite 60° would be the long leg. Wait, the right angle, angle 60°, so the side adjacent to 60° (the leg) is the short leg? No, wait, the angle of 30°: the side opposite 30° is the short leg. So if the angle is 30°, the side opposite is short leg. Let's look at the triangle: the side labeled 6 is adjacent to the 60° angle and opposite the 30° angle? Wait, no, the right angle is at the top left, the 60° angle is at the bottom left, so the side with length 6 is the leg from the right angle to the 60° angle, so that's adjacent to 60°, so it's opposite the 30° angle. So that's the short leg (\(a = 6\))? Wait, no, wait: if the angle is 30°, the side opposite is short leg. So if the side of length 6 is opposite 30°, then:

- Short leg (\(a\)) = 6? Wait, no, that can't be, because then the hypotenuse would be \(2a = 12\), and the long leg (opposite 60°) would be \(6\sqrt{3}\). But wait, the side labeled \(x\) is the long leg (opposite 60°), and \(y\) is the hypotenuse? Wait, no, let's check the triangle again. The right angle is at the top left, so the legs are: horizontal leg \(x\), vertical leg 6, hypotenuse \(y\). The angle at the bottom left is 60°, so the angle between the vertical leg (6) and hypotenuse (\(y\)) is 60°, so the vertical leg (6) is adjacent to 60°, so it's opposite the 30° angle (at the top right). So the short leg (opposite 30°) is 6. Then:

- Short leg (\(a\)) = 6
- Long leg (opposite 60°) = \(a\sqrt{3} = 6\sqrt{3}\) (this is \(x\))
- Hypotenuse (\(y\)) = \(2a = 12\)

Wait, but let's confirm. The angle at the bottom left is 60°, so the angle at the top right is 30°, so the side opposite 30° is the vertical leg (6), so that's the short leg. Then the horizontal leg (\(x\)) is opposite 60°, so it's the long leg, length \(6\sqrt{3}\), and hypotenuse \(y = 12\).

Wait, but maybe I mixed up the legs. Let's use trigonometry ratios (even though the problem says not to use trig tool, but we can use the 30-60-90 ratios). So in a right triangle, with angle 60°, the sides:

- Adjacent to 60°: 6 (vertical leg)
- Opposite to 60°: \(x\) (horizontal leg)
- Hypotenuse: \(y\)

In 30-60-90, the side adjacent to 60° (which is the short leg, opposite 30°) is \(a\), so \(a = 6\). Then:

- Opposite 60° (long leg) = \(a\sqrt{3} = 6\sqrt{3}\) (so \(x = 6\sqrt{3}\))
- Hypotenuse = \(2a = 12\) (so \(y = 12\))

Wait, but let's check with angles. If the short leg is 6 (opposite 30°), then hypotenuse is 12, long leg is \(6\sqrt{3}\). That makes sense.

Step2: Calculate \(x\) and \(y\)

- \(x\) is the long leg (opposite 60°), so \(x = 6\sqrt{3}\)
- \(y\) is the hypotenuse, so \(y = 2 \times 6 = 12\)

Wait, but let's confirm the angle. The angle at the bottom left is 60°, so the side adjacent to 60° is 6 (vertical leg), which is opposite 30°, so that's the short leg. Therefore, hypotenuse is twice the short leg, so \(y = 12\), and the long leg (opposite 60°) is \(6\sqrt{3}\), which is \(x\).

Answer:

\(x = 6\sqrt{3}\), \(y = 12\)