If $|1 + 2|z^2| = |z^2 + 1| + |2|z| + 1|^2$, then the value of $\frac{|z(z+1)|}{2}$ is ____.
Step-by-Step Solution
Key Concept: Parallelograms are formed by choosing 2 lines from one set and 2 lines from another set of parallel lines; the total count is the square of combinations.
Given $m$ parallel lines, two additional lines can be selected from $m$ lines in $C_m^2$ ways. Since each pair of lines determines a parallelogram with another pair of lines, the total number of parallelograms formed is $(C_m^2)^2 = 225$. This means $C_m^2 = 15$, giving $m(m-1)/2 = 15$, so $m = 6$.
Correct Answer: 0.50