https://file.aiquickdraw.com/mathbot/file/17273276409157w7bgx1o.jpg

We have a right triangle in the image, where:

- One angle is \( 30^\circ \),
- The side opposite the \( 30^\circ \) angle is 6 units.

To find the other sides of the triangle, we can use the properties of a 30-60-90 triangle. In such triangles, the sides have a consistent ratio:

- The side opposite the \( 30^\circ \) angle is \( x \),
- The side opposite the \( 60^\circ \) angle is \( x \sqrt{3} \),
- The hypotenuse is \( 2x \).

Here, the side opposite the \( 30^\circ \) angle is given as 6, so we set \( x = 6 \).

Now, we can calculate the other sides:

1. **The hypotenuse**:  
   \[
   \text{Hypotenuse} = 2x = 2 \times 6 = 12
   \]

2. **The side opposite the \( 60^\circ \) angle**:  
   \[
   \text{Side opposite } 60^\circ = x \sqrt{3} = 6 \sqrt{3}
   \]

So, the lengths of the sides of the triangle are:

- Side opposite the \( 30^\circ \) angle: \( 6 \) units,
- Side opposite the \( 60^\circ \) angle: \( 6 \sqrt{3} \) units,
- Hypotenuse: \( 12 \) units.

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions you might find useful:

1. How do you derive the ratio of sides in a 30-60-90 triangle?
2. How can you find the angles of a triangle given all sides?
3. What is the sine, cosine, and tangent of a \( 30^\circ \) angle in a right triangle?
4. How can you use the Pythagorean theorem to verify the side lengths in a right triangle?
5. How does the ratio of sides change in other special triangles, like the 45-45-90 triangle?

**Tip**: In a 30-60-90 triangle, always remember the side opposite the smallest angle (30°) is half the hypotenuse!

Learn how to find the sides of a 30-60-90 triangle, where one side is given as 6 units. Using special right triangle properties, calculate the other sides in fractional form.

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