Question

use the information and diagram to answer the following question. the side of a square is \\(\sqrt{8}\\) units as shown in the diagram. which of the following statements is true? (a) the perimeter of the square is 8 units, and 8 is an irrational number. (b) the perimeter of the square is \\(4\sqrt{8}\\) units, and \\(4\sqrt{8}\\) is an irrational number. (c) the perimeter of the square is 8 units, and 8 is a rational number. (d) the perimeter of the square is \\(4\sqrt{8}\\) units, and \\(4\sqrt{8}\\) is a rational number.

Explanation:

Step1: Recall Square Perimeter Formula

The perimeter \( P \) of a square with side length \( s \) is \( P = 4s \). Here, \( s=\sqrt{8} \), so \( P = 4\sqrt{8} \).

Step2: Classify \( 4\sqrt{8} \) as Rational/Irrational

A rational number can be expressed as \( \frac{p}{q} \) ( \( p,q \) integers, \( q
eq0 \) ), but \( \sqrt{8}=2\sqrt{2} \), so \( 4\sqrt{8}=8\sqrt{2} \), which is irrational (since \( \sqrt{2} \) is irrational and multiplying by a non - zero rational number keeps it irrational). Now check options:

• Option A: Perimeter calculation wrong (should be \( 4\sqrt{8} \), not 8), eliminate.
• Option B: Perimeter is \( 4\sqrt{8} \), and \( 4\sqrt{8} \) is irrational (as shown), correct.
• Option C: Perimeter wrong and 8 is rational, but perimeter is not 8, eliminate.
• Option D: \( 4\sqrt{8} \) is irrational, not rational, eliminate.

Answer:

B. The perimeter of the square is \( 4\sqrt{8} \) units, and \( 4\sqrt{8} \) is an irrational number