https://file.aiquickdraw.com/mathbot/file/1730469615228gbw9jd9x.jpg

The question asks for the values of \( x \), \( y \), and \( z \) in the triangles shown. Let's analyze each triangle:

### Analyzing Triangle \( \triangle ABC \) on the Left:
1. Triangle \( \triangle ABC \) is a right triangle (note the \(90^\circ\) angle at \( D \)).
2. Given:
   - \( \angle B = 60^\circ \)
   - \( \angle C = 30^\circ \)
   - Hypotenuse \( AC = 12 \)

Using trigonometric ratios in a \( 30^\circ - 60^\circ - 90^\circ \) triangle:
- The side opposite \(30^\circ\) (here \( BD \), denoted \( y \)) is half the hypotenuse:
  \[
  y = \frac{AC}{2} = \frac{12}{2} = 6
  \]
- The side opposite \(60^\circ\) (here \( AB \), denoted \( z \)) is \( \frac{\sqrt{3}}{2} \) times the hypotenuse:
  \[
  z = \frac{\sqrt{3}}{2} \cdot 12 = 6\sqrt{3}
  \]
- The side \( x \) (adjacent to \( 30^\circ \)) is equal to the side opposite \(60^\circ\):
  \[
  x = BD = 6\sqrt{3}
  \]

### Solution:
- \( x = 6\sqrt{3} \)
- \( y = 6 \)
- \( z = 6\sqrt{3} \)

Let me know if you would like more details on these calculations or if you have any questions.

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### Related Questions:
1. What are the properties of a \(30^\circ - 60^\circ - 90^\circ\) triangle?
2. How do trigonometric ratios help in solving right triangles?
3. What is the role of similar triangles in calculating side lengths?
4. How is the Pythagorean theorem used in right triangles?
5. How do angles in a triangle determine the length of the sides?

**Tip:** In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the ratio of the side lengths is always \(1 : \sqrt{3} : 2\).

در شکل زیر، مقدار x، y و z را حساب کنید.

Calculate the values of x, y, and z in a 30-60-90 triangle where the hypotenuse is given as 12. This problem demonstrates the application of trigonometric ratios and the properties of special right triangles, including the Pythagorean theorem.

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