Question

A square with a side length of s inches has a perimeter that is rational. Which of the following could be the length of s?

square root of 12
square root of 20
2square root of 5
3square root of 9

Answer 1

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.

1. **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.

2. **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.

3. **Option 3: \( 2\sqrt{5} \)**
\( \sqrt{5} \) is irrational, thus \( 2\sqrt{5} \) is also irrational. Hence, \( s = 2\sqrt{5} \) is irrational.

4. **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:

\[
\boxed{3\sqrt{9}}
\]