1. **State the problem:** We are given a complex number $z$ that satisfies the equation $$z + i|z| = 12 + 9i.$$ We need to find the imaginary part of $z$. 2. **Express $z$ in terms of its real and imaginary parts:** Let $$z = x + yi,$$ where $x$ and $y$ are real numbers, and $i$ is the imaginary unit. 3. **Recall the modulus of $z$:** The modulus (or absolute value) of $z$ is $$|z| = \sqrt{x^2 + y^2}.$$ This is always a non-negative real number. 4. **Rewrite the given equation using $x$ and $y$:** Substitute $z = x + yi$ and $|z| = \sqrt{x^2 + y^2}$ into the equation: $$x + yi + i\sqrt{x^2 + y^2} = 12 + 9i.$$ 5. **Separate real and imaginary parts:** The left side real part is $x$, and the imaginary part is $$y + \sqrt{x^2 + y^2}.$$ The right side real part is 12, and the imaginary part is 9. So, we have the system: $$\begin{cases} x = 12 \\ y + \sqrt{12^2 + y^2} = 9 \end{cases}$$ 6. **Solve for $y$:** From the second equation, $$y + \sqrt{144 + y^2} = 9.$$ Isolate the square root: $$\sqrt{144 + y^2} = 9 - y.$$ Since the square root is non-negative, we require $9 - y \geq 0 \Rightarrow y \leq 9$. 7. **Square both sides:** $$144 + y^2 = (9 - y)^2 = 81 - 18y + y^2.$$ Subtract $y^2$ from both sides: $$144 = 81 - 18y.$$ 8. **Solve for $y$:** $$144 - 81 = -18y \Rightarrow 63 = -18y \Rightarrow y = -\frac{63}{18} = -\frac{7}{2} = -3.5.$$ 9. **Check the condition $y \leq 9$:** $-3.5 \leq 9$ is true, so this solution is valid. 10. **Final answer:** The imaginary part of $z$ is $$\boxed{-3.5}.$$