To solve the equation \((x-3)^{2} = (2x-2)^{2}\), we can take the square root of both sides. However, remember to consider both the positive and negative roots.

### Step 1: Take Square Roots
\[
x - 3 = 2x - 2 \quad \text{or} \quad x - 3 = -(2x - 2)
\]

### Step 2: Solve the First Equation
1. \(x - 3 = 2x - 2\):
\[
-3 + 2 = 2x - x \\
-1 = x \\
x = -1
\]

### Step 3: Solve the Second Equation
2. \(x - 3 = -2x + 2\):
\[
x + 2x = 2 + 3 \\
3x = 5 \\
x = \frac{5}{3}
\]

### Step 4: Final Solutions
The solutions to the equation \((x-3)^{2} = (2x-2)^{2}\) are:
\[
x = -1 \quad \text{and} \quad x = \frac{5}{3}
\]