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To create a dynamic geometry board for question 22, I'll first describe the problem in detail and then outline the steps for creating the board.

### Problem Description

In triangle \( \triangle ABC \):
- \( AC = 4 \)
- \( \angle BAC = 60^\circ \)
- \( \angle ABC = 45^\circ \)
- \( AD \) is the altitude from \( A \) to \( BC \)
- \( P \) is a point on \( AB \)
- \( Q \) is a point on \( BC \)

The tasks are:
1. Find the length of \( AD \).
2. \( P \) moves along \( AB \), and using \( DP \), construct parallelogram \( DPQB \).
3. Find the minimum length of \( CQ \).

### Dynamic Geometry Board

To create a dynamic geometry board, use software like GeoGebra. Here are the steps to create it:

1. **Create Triangle \( \triangle ABC \):**
   - Place points \( A \), \( B \), and \( C \) with given side lengths and angles.
   - Ensure \( AC = 4 \), \( \angle BAC = 60^\circ \), and \( \angle ABC = 45^\circ \).

2. **Construct Altitude \( AD \):**
   - Draw a perpendicular line from \( A \) to \( BC \) to locate point \( D \).

3. **Dynamic Point \( P \):**
   - Place a point \( P \) on segment \( AB \).
   - Make \( P \) movable along \( AB \).

4. **Construct Parallelogram \( DPQB \):**
   - Using the dynamic point \( P \) on \( AB \), construct segment \( DP \).
   - Ensure \( DP \) is parallel to \( BQ \) and \( DQ \) is parallel to \( BP \) to form parallelogram \( DPQB \).

5. **Track the Length of \( CQ \):**
   - Calculate the length of \( CQ \) as \( P \) moves along \( AB \).
   - Add functionality to display the length of \( CQ \) dynamically.

By using these steps in GeoGebra, you can visualize and manipulate the geometry to understand the problem better and find the minimum length of \( CQ \).

### Additional Help

If you need further assistance with using GeoGebra or any other dynamic geometry software, feel free to ask. Here's a list of questions you might want to explore next:

1. How to find the coordinates of specific points in a triangle given side lengths and angles?
2. How to derive the equations of lines given specific points or angles?
3. How to use trigonometric identities to solve geometric problems?
4. How to calculate the area of parallelograms in various geometric configurations?
5. How to determine the properties of special quadrilaterals formed by specific points in a triangle?

### Tip

When working with dynamic geometry software, always start with the basic geometric construction and incrementally add constraints and dynamic elements to ensure accuracy and ease of manipulation.

Learn how to create a dynamic geometry board for problem 22 involving triangle ABC. This problem covers constructing triangles, calculating altitudes, and forming parallelograms using dynamic points. Suitable for high school students.

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