Perimeter of a square is 24 $\sqrt{2}$ cm. Its diagonal is :

Question

Accepted Answer

To determine the diagonal of a square when the perimeter is given, we can follow these steps:

1. **Understanding the Perimeter of a Square:**
   - The perimeter of a square is given by the formula $ P = 4s $, where $ s $ is the side length of the square.
   - We are given that the perimeter is $ 24\sqrt{2} $ cm. Therefore, we can set up the equation:
     $$
     4s = 24\sqrt{2}
     $$
   - Solving for $ s $, we divide both sides by 4:
     $$
     s = \frac{24\sqrt{2}}{4} = 6\sqrt{2} \ \text{cm}
     $$

2. **Finding the Diagonal of the Square:**
   - The diagonal $ d $ of a square can be found using the formula $ d = s\sqrt{2} $.
   - Substituting the value of $ s $ we found:
     $$
     d = 6\sqrt{2} \times \sqrt{2} = 6 \times 2 = 12 \ \text{cm}
     $$

3. **Justification for Option 4:**
   - The calculated diagonal of the square is 12 cm, which matches Option 4. Therefore, Option 4 is the correct answer.

4. **Explanation of Why Other Options are Incorrect:**
   - **Option 1: $ 6\sqrt{2} \ \text{cm} $**: This option incorrectly assumes the diagonal is equal to the side length, which is not the case for a square.
   - **Option 2: $ 8\sqrt{2} \ \text{cm} $**: This option does not match the calculated diagonal based on the given perimeter.
   - **Option 3: 8 cm**: This option is incorrect as it does not account for the relationship between the side length and the diagonal of a square.

In conclusion, based on the given perimeter and the calculations, the diagonal of the square is indeed 12 cm, which supports Option 4 as the correct answer.