https://file.aiquickdraw.com/mathbot/file/1725637352407f43g1mnn.jpg

To solve the problem, let's analyze the figure and the given information:

### Problem Breakdown:
1. **Figure Description**:
   - \( ADEF \) is a rectangle.
   - \( OABC \) is a square.
   - Lines \( FD \) and \( AE \) intersect at \( O \).
   - \( AF = x \), \( AE = y \), and \( ED = z \).

2. **Given Conditions**:
   - \( FD \) and \( AE \) intersect at \( O \).
   - \( AF = OF \) (since \( OABC \) is a square, \( O \) is the center, and diagonals of a square bisect each other).

### Steps to Solve:
1. **Coordinates and Relationships**:
   - Let the square \( OABC \) have side length \( a \).
   - \( AF = x \), \( AE = y \), and \( ED = z \).

   Since \( OABC \) is a square and \( O \) is the center, the coordinates of \( O \) are the midpoint of diagonals \( AC \) and \( BD \).
   
   Because \( ADEF \) is a rectangle and \( OABC \) is a square, the diagonal properties will help us relate \( x \), \( y \), and \( z \) using the Pythagorean theorem.

2. **Apply Geometric Relationships**:
   - Since \( OABC \) is a square, \( OA = OB = OC = OD = \frac{\sqrt{2}}{2}a \).
   - \( AF = x \) and \( AE = y \) implies that triangle \( OAF \) is isosceles, so:
     \[
     AF = FO = x
     \]
     And since \( ADEF \) is a rectangle:
     \[
     AE = ED = y = z
     \]
     This implies:
     \[
     x^2 + y^2 = z^2
     \]
   - Given \( FD \) intersects \( AE \) at \( O \), we have an additional relationship \( x + z = 2a \).

### Solution:
   Now, solving the system of equations:
   \[
   x^2 + y^2 = z^2
   \]
   and
   \[
   x + z = 2a
   \]

   **The specific values** of \( x \), \( y \), and \( z \) would depend on solving these equations given the specific measures or deriving them from the geometric properties of the square and rectangle.

Let me know if you'd like a more detailed calculation or explanation on a specific part. Would you like the exact coordinates and detailed steps, or any clarification?

### Relative Questions:
1. How would the answer change if the square were not centered on the rectangle?
2. What is the effect of increasing the size of the square on the lengths \( x \), \( y \), and \( z \)?
3. How do properties of rectangles and squares influence the relationships between \( x \), \( y \), and \( z \)?
4. Can you derive another geometric figure with similar properties using different starting shapes?
5. What would happen if \( OABC \) were not a square but a different quadrilateral?

### Tip:
When solving geometric problems involving squares and rectangles, remember to use symmetry and the properties of diagonals effectively to simplify the problem.

In the figure, ADEF is a rectangle and OABC is a square. It is given that FD and AE intersect at O, and AF = OF. Find the values of x, y, and z.

Solve a geometric problem where ADEF is a rectangle and OABC is a square. FD and AE intersect at O. Use the properties of diagonals and Pythagorean theorem to find the values of x, y, and z. Suitable for students in Grades 9-12.

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