Step 1: Understanding the Concept:
This problem requires performing arithmetic operations with complex numbers, including squaring a complex number and dividing complex numbers. The final step is to identify the real part of the resulting complex number.
Step 2: Key Formula or Approach:
1. Expand the denominator: $(a-b)^2 = a^2 - 2ab + b^2$. Remember that $i^2 = -1$.
2. Divide complex numbers: To divide $\frac{z_1}{z_2}$, multiply the numerator and denominator by the conjugate of the denominator, $\bar{z_2}$.
Step 3: Detailed Explanation:
Let the given complex number be $z = \frac{1+2i}{(2-i)^2}$.
First, simplify the denominator:
\[ (2-i)^2 = 2^2 - 2(2)(i) + i^2 = 4 - 4i - 1 = 3 - 4i \]
So, the expression becomes:
\[ z = \frac{1+2i}{3-4i} \]
To simplify this fraction, multiply the numerator and denominator by the conjugate of the denominator, which is $3+4i$:
\[ z = \frac{1+2i}{3-4i} \times \frac{3+4i}{3+4i} \]
Multiply the numerators:
\[ (1+2i)(3+4i) = 1(3) + 1(4i) + 2i(3) + 2i(4i) = 3 + 4i + 6i + 8i^2 = 3 + 10i - 8 = -5 + 10i \]
Multiply the denominators (using the property $(a-bi)(a+bi) = a^2+b^2$):
\[ (3-4i)(3+4i) = 3^2 + 4^2 = 9 + 16 = 25 \]
So, the complex number is:
\[ z = \frac{-5 + 10i}{25} = \frac{-5}{25} + \frac{10i}{25} = -\frac{1}{5} + \frac{2}{5}i \]
The real part of a complex number $a+bi$ is $a$.
In this case, the real part is $-\frac{1}{5}$.
Step 4: Final Answer:
The real part of the given complex number is $-\frac{1}{5}$. Therefore, option (A) is correct.