30-60-90 Triangle with side ratios 1:√3:2.

The image shows a 30-60-90 triangle, which is a special right triangle with angles measuring 30°, 60°, and 90°. The sides of this triangle follow a specific ratio:

$$
1 : \sqrt{3} : 2
$$

This means:
- The side opposite the 30° angle is the shortest side: $ x $
- The side opposite the 60° angle is $ x\sqrt{3} $
- The hypotenuse (opposite the 90° angle) is $ 2x $

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Let's analyze the triangle in the image:

- The right angle is at the bottom-left corner.
- The 30° angle is at the top.
- The 60° angle is at the bottom-right.
- The side opposite the 30° angle is labeled $ x $
- The side opposite the 60° angle is labeled $ x\sqrt{3} $
- The hypotenuse is labeled $ 2x $

This matches the standard 30-60-90 triangle ratios perfectly.

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Key Properties of a 30-60-90 Triangle:

| Angle | Side Opposite |
|-------|----------------|
| 30° | $ x $ |
| 60° | $ x\sqrt{3} $ |
| 90° | $ 2x $ |

So if you know one side, you can find the other two using this ratio.

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Example: Suppose $ x = 5 $

Then:
- Side opposite 30°: $ x = 5 $
- Side opposite 60°: $ x\sqrt{3} = 5\sqrt{3} $
- Hypotenuse: $ 2x = 10 $

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Summary:

✔ The triangle shown is a classic 30-60-90 triangle.
✔ The side lengths are in the ratio $ 1 : \sqrt{3} : 2 $, where:
- $ x $ is the side opposite 30°,
- $ x\sqrt{3} $ is the side opposite 60°,
- $ 2x $ is the hypotenuse.

This relationship is useful for solving problems involving right triangles with these angles without needing trigonometry.

Let me know if you'd like to solve a specific problem using this triangle!