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The given equation is:

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

Where:
- \( z \) is a complex number, so we can express it as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part.
- \( \overline{z} \) is the complex conjugate of \( z \), so \( \overline{z} = x - yi \).
- \( |z| \) is the magnitude of \( z \), which is \( |z| = \sqrt{x^2 + y^2} \).

### Step-by-step solution:
1. Substitute \( \overline{z} = x - yi \) and \( |z| = \sqrt{x^2 + y^2} \) into the equation:
   \[
   (x - yi) + i\sqrt{x^2 + y^2} = 12 + 9i
   \]

2. Now, equate both the real and imaginary parts from both sides:
   - Real part: \( x = 12 \)
   - Imaginary part: \( -y + \sqrt{x^2 + y^2} = 9 \)

3. Substitute \( x = 12 \) into the imaginary part equation:
   \[
   -y + \sqrt{12^2 + y^2} = 9
   \]
   Simplify:
   \[
   -y + \sqrt{144 + y^2} = 9
   \]

4. Solve for \( y \):
   - First, isolate the square root term:
     \[
     \sqrt{144 + y^2} = 9 + y
     \]
   - Square both sides:
     \[
     144 + y^2 = (9 + y)^2
     \]
     Expand the right-hand side:
     \[
     144 + y^2 = 81 + 18y + y^2
     \]
   - Cancel out \( y^2 \) on both sides:
     \[
     144 = 81 + 18y
     \]
   - Solve for \( y \):
     \[
     63 = 18y \quad \Rightarrow \quad y = \frac{63}{18} = 3.5
     \]

### Answer:
The imaginary part of \( z \) is \( y = 3.5 \).

Would you like further details or clarifications on any step?

Here are 5 questions you can explore:
1. How do we compute the magnitude \( |z| \) of a complex number?
2. What is the significance of the complex conjugate in this equation?
3. How can complex numbers be represented geometrically?
4. Can you find an alternative method to solve this type of equation?
5. How does the relationship between the real and imaginary parts affect the solution?

**Tip:** In complex numbers, always separate real and imaginary parts to simplify 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?

Learn how to find the imaginary part of a complex number z using the equation \overline{z} + i|z| = 12 + 9i. This problem explores key algebraic concepts like complex conjugates and magnitudes, suitable for students in Grades 10-12.

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