To determine the largest possible area of a square that can be made from the pieces cut from the three steel bars, we first need to find the greatest common divisor (GCD) of the lengths of the bars. When we cut the bars into equal pieces, each piece will have a length equal to the GCD.
The lengths of the bars are:
- 200 cm
- 300 cm
- 360 cm
Step 1: Finding the GCD of the lengths.
To find the GCD, we can use the prime factorization method.
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Prime factorization of each length:
- \( 200 = 2^3 \times 5^2 \)
- \( 300 = 2^2 \times 3^1 \times 5^2 \)
- \( 360 = 2^3 \times 3^2 \times 5^1 \)
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Taking the minimum power for each prime factor:
- For \(2\): Minimum power is \(2\) (from \(300\)).
- For \(3\): Minimum power is \(0\) (from \(200\)).
- For \(5\): Minimum power is \(1\) (from \(360\)).
Thus,
\[ \text{GCD} = 2^2 \times 3^0 \times 5^1 = 4 \times 1 \times 5 = 20. \]
Step 2: Calculating the area of the square.
The length of each piece is 20 cm. The largest square that can be made from one of the pieces will have a side length equal to the length of the piece, which is 20 cm.
Now, we calculate the area \(A\) of the square:
\[ A = \text{side}^2 = 20^2 = 400 \text{ cm}^2. \]
Thus, the largest possible area of a square that can be made from any of the pieces is:
\[ \boxed{400} \text{ cm}^2. \]
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