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The triangle in the image is a 30°-60°-90° right triangle. For this type of triangle, the sides have a known ratio:

- The side opposite the 30° angle is \( \frac{1}{2} \) of the hypotenuse.
- The side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
- The hypotenuse is the longest side, denoted by \( y \) in this case.

### Step-by-step approach:

1. The side labeled \( 6 \) is opposite the 60° angle, so this is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
2. Let \( y \) represent the hypotenuse, then we can write:
   \[
   6 = \frac{\sqrt{3}}{2} y
   \]
3. Solve for \( y \) (the hypotenuse):
   \[
   y = \frac{6 \times 2}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}
   \]
4. Now, for the side opposite the 30° angle (labeled \( x \)), we know it is \( \frac{1}{2} \) of the hypotenuse:
   \[
   x = \frac{1}{2} y = \frac{1}{2} \times 4\sqrt{3} = 2\sqrt{3}
   \]

Thus:
- The hypotenuse \( y = 4\sqrt{3} \).
- The side opposite the 30° angle \( x = 2\sqrt{3} \).

Would you like further clarification or additional details on any step?

Here are 5 related questions for further practice:
1. How would the solution change if the side opposite the 60° angle was 10 instead of 6?
2. What is the area of the triangle based on the calculated side lengths?
3. How do you find the perimeter of the triangle using the values of \( x \) and \( y \)?
4. What trigonometric identities can be used to verify the side lengths in this triangle?
5. How would the side lengths change in a 45°-45°-90° triangle?

**Tip**: Remember that 30°-60°-90° triangles always maintain a consistent ratio of side lengths: 1 : \( \sqrt{3} \) : 2!

Solve the triangle with a 30°-60°-90° configuration. One side opposite the 60° angle is 6 units. Find the hypotenuse and the side opposite the 30° angle.

Learn how to solve a 30°-60°-90° triangle where the side opposite the 60° angle is given as 6 units. Calculate the hypotenuse and the side opposite the 30° angle using special triangle properties.

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