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
?

(1 point)
Responses

20−−√
square root of 20

39–√
3 square root of 9

12−−√
square root of 12

25–√
2 square root of 5

Answer 1

To determine which of the given lengths \(s\) would result in a rational perimeter for a square, we start with the formula for the perimeter \(P\) of a square:

\[
P = 4s
\]

The perimeter is rational if \(s\) is rational. Let's evaluate each of the given options for \(s\):

1. **\(\sqrt{20}\)**:
\[
\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}
\]
This is an irrational number.

2. **\(3\sqrt{9}\)**:
\[
3\sqrt{9} = 3 \times 3 = 9
\]
This is a rational number.

3. **\(\sqrt{12}\)**:
\[
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]
This is an irrational number.

4. **\(2\sqrt{5}\)**:
This is already stated as \(s\) being \(2\sqrt{5}\), which is an irrational number.

Thus, among the given options, the only length that results in a rational perimeter when the side length \(s\) is \(3\sqrt{9} = 9\).

Therefore, the correct answer is:
**\(3\sqrt{9}\)**