To determine which of the given options for \( s \) results in a rational perimeter for the square, we first need to recall the formula for the perimeter \( P \) of a square, which is given by:
\[ P = 4s \]
The perimeter will be rational if \( s \) itself is a rational number. We will analyze each of the options one by one to see if \( s \) is rational.
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Option 1: \( \sqrt{12} \) \[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \] Here, \( \sqrt{3} \) is irrational, so \( 2\sqrt{3} \) is also irrational. Hence, \( s = \sqrt{12} \) is irrational.
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Option 2: \( \sqrt{20} \) \[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \] Similarly, \( \sqrt{5} \) is irrational, so \( 2\sqrt{5} \) is also irrational. Hence, \( s = \sqrt{20} \) is irrational.
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Option 3: \( 2\sqrt{5} \) \( \sqrt{5} \) is irrational, thus \( 2\sqrt{5} \) is also irrational. Hence, \( s = 2\sqrt{5} \) is irrational.
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Option 4: \( 3\sqrt{9} \) \[ \sqrt{9} = 3 \implies 3\sqrt{9} = 3 \times 3 = 9 \] In this case, \( 9 \) is a rational number. Hence, \( s = 3\sqrt{9} \) (which equals 9) is rational.
Now that we have analyzed each option, we find only one option where \( s \) is a rational number:
- Option 4: \( 3\sqrt{9} \) (which equals 9)
Thus, the length of \( s \) that results in a rational perimeter is \( 3\sqrt{9} \). The perimeter in this case would be \( P = 4 \times 9 = 36 \), which is rational.
The correct answer is:
\[ \boxed{3\sqrt{9}} \]