Question

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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
?

(1 point)
Responses

39–√
3 square root of 9

12−−√
square root of 12

25–√
2 square root of 5

20−−√
square root of 20
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Answer 1

The perimeter of a square is given by the formula \( P = 4s \), where \( s \) is the side length of the square. For the perimeter to be rational, \( s \) must be such that \( 4s \) is a rational number.

Let's analyze each option:

1. **\(39 - \sqrt{3}\)**: This is a rational number because it consists of a rational number (39) minus an irrational number (\(\sqrt{3}\)). The result is irrational.

2. **\( \sqrt{9} \)**: This simplifies to \( 3 \), which is rational. Therefore, \( 4s = 4 \times 3 = 12 \), which is rational.

3. **\( \sqrt{12} \)**: This simplifies to \( 2\sqrt{3} \), which is irrational. Thus, \( 4s \) would be \( 4 \times 2\sqrt{3} = 8\sqrt{3} \), which is also irrational.

4. **\( \sqrt{5} \)**: This is irrational. Hence, \( 4s = 4\sqrt{5} \), which is irrational.

5. **\( \sqrt{20} \)**: This simplifies to \( 2\sqrt{5} \), which is irrational. So, \( 4s = 8\sqrt{5} \), which is also irrational.

The only option where the side length \( s \) leads to a rational perimeter is **\( \sqrt{9} \)**, which equals **3**.

Thus, the correct answer is **\( \sqrt{9} \)**.