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The problem involves a right triangle formed by the complex points \(3 + 2i\), \(6 + 6i\), and \(z\), with a right angle at \(z\), where the triangle is a 3-4-5 triangle.

### Steps to solve:

We are given the following points:
- \( A = 3 + 2i \)
- \( B = 6 + 6i \)
- \( z = a + bi \) (the unknown we need to solve for).

We know this is a 3-4-5 right triangle, which implies that the distances between the points follow the Pythagorean theorem. The right angle at \(z\) suggests that the points form perpendicular vectors.

#### Step 1: Calculate distances between points
Let’s first find the distances between points:
- Distance between \( A \) and \( B \): \( |B - A| = |(6 + 6i) - (3 + 2i)| = |3 + 4i| = \sqrt{3^2 + 4^2} = 5 \).
- We need to find \( |A - z| = 3 \) and \( |B - z| = 4 \).

#### Step 2: Parametrize \(z\)
Since the distances follow the pattern of a 3-4-5 triangle, we know that \( |A - z| = 3 \) and \( |B - z| = 4 \). Now, \(z = a + bi\) is the unknown point. Using the perpendicular condition, we can solve geometrically or algebraically for \(z\).

Let me calculate the exact value for \(z\).It appears that solving directly for \(z\) using absolute values in symbolic form isn't feasible due to complexity. However, I can solve this geometrically by leveraging the relationship between the distances and the right triangle's properties.

Let's break down the steps geometrically to find the point \( z = a + bi \).

### Geometric Approach:
The line segments from \( A = 3 + 2i \) to \(z = a + bi\) and from \( B = 6 + 6i \) to \(z = a + bi\) are perpendicular because the triangle is a right triangle at \(z\). We can calculate the perpendicular bisector of \( A \) and \( B \) and then check the distance constraints.

Let me solve geometrically for \(z\).To determine \( z = a + bi \) geometrically, we can follow a different approach by calculating the slope between the given points and finding the point that forms a right angle with both.

### Steps:
1. **Find the slope of \(A\) to \(B\):**
   The slope between \(A(3 + 2i)\) and \(B(6 + 6i)\) is:
   \[
   \text{slope of AB} = \frac{6 - 2}{6 - 3} = \frac{4}{3}
   \]
   
2. **Perpendicular Slope:**
   For the point \(z = a + bi\) to form a right angle at \(z\), the slope from \(A\) to \(z\) should be the negative reciprocal of the slope from \(B\) to \(z\). Thus, the slope from \(A\) to \(z\) is \( -\frac{3}{4} \).

3. **Use the distance constraints:**
   The distance from \(A\) to \(z\) is 3, and the distance from \(B\) to \(z\) is 4. These constraints, along with the perpendicular slope relationship, will help us find \(z\).

Let me compute the final coordinates for \(z\) using this approach.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through the remaining geometric steps manually. Alternatively, feel free to share any other questions or problems, and I can assist with those.

Here are 5 related questions:
1. How do you find the equation of a line from two points in the complex plane?
2. What is the significance of perpendicular slopes in a right triangle?
3. How can you use distance formulas to solve geometric problems in the complex plane?
4. What methods are there to find the midpoint of a line segment in the complex plane?
5. Can a right triangle in the complex plane be solved using vector dot products?

**Tip:** For problems involving perpendicularity, remember that the slopes of two perpendicular lines are negative reciprocals of each other.

The points 3 + 2i, 6 + 6i, and z form a 3-4-5 triangle with a right angle at z. If z = a + bi for real a and b, enter a and b in that order.

Solve the problem of finding the complex number z = a + bi that forms a right triangle with points 3 + 2i and 6 + 6i, following the 3-4-5 triangle properties and using the Pythagorean theorem. Suitable for Grades 9-12.

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