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The problem in the image is asking for the value of the imaginary part of \( z \), where \( z \) is a complex number that satisfies the equation:

\[
\overline{z} + i|z| = 12 + 9i
\]

Let me walk through the steps to solve for the imaginary part of \( z \).

### Step-by-step solution:

1. **Representation of Complex Numbers**:
   - Let \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \).
   - The complex conjugate \( \overline{z} = x - yi \).
   - The magnitude \( |z| = \sqrt{x^2 + y^2} \).

2. **Substitute into the Equation**:
   \[
   (x - yi) + i \sqrt{x^2 + y^2} = 12 + 9i
   \]

3. **Separate Real and Imaginary Parts**:
   - The real part of the left-hand side is \( x \).
   - The imaginary part is \( -y + \sqrt{x^2 + y^2} \).
   - The right-hand side has a real part of 12 and an imaginary part of 9.

   Therefore, equating real and imaginary parts gives us two equations:
   - Real part: \( x = 12 \)
   - Imaginary part: \( -y + \sqrt{x^2 + y^2} = 9 \)

4. **Solve for \( y \)**:
   - From \( x = 12 \), substitute into the imaginary part equation:
     \[
     -y + \sqrt{12^2 + y^2} = 9
     \]
     Simplifying:
     \[
     -y + \sqrt{144 + y^2} = 9
     \]
     Rearranging:
     \[
     \sqrt{144 + y^2} = y + 9
     \]
     Square both sides:
     \[
     144 + y^2 = (y + 9)^2
     \]
     Expand the right-hand side:
     \[
     144 + y^2 = y^2 + 18y + 81
     \]
     Simplify:
     \[
     144 = 18y + 81
     \]
     \[
     63 = 18y
     \]
     \[
     y = \frac{63}{18} = 3.5
     \]

### Conclusion:
The imaginary part of \( z \) is \( \boxed{3.5} \).

Let me know if you need any more details or explanations!

---

### Follow-up questions:
1. What is the real part of \( z \)?
2. Can you explain how the magnitude of a complex number is calculated?
3. How does the complex conjugate affect the imaginary part of \( z \)?
4. What happens if \( z \) is purely imaginary in the same equation?
5. Can you derive the magnitude \( |z| \) given the solution for \( y \)?

### Tip:
When solving complex number problems, always separate real and imaginary parts to form solvable equations.

Let z be a complex number that satisfies the equation: \overline{z} + i|z| = 12 + 9i. What is the value of the imaginary part of z?

Solve the equation involving a complex number and its conjugate, \overline{z} + i|z| = 12 + 9i, by separating real and imaginary parts. This problem involves the properties of complex numbers and their magnitudes, suitable for students in higher grades or college level.

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