The diagonal of a square is $4\sqrt{2}cm$. The diagonal of another square whose area is double that of the first square, is:

8 cm

The correct answer is Option (4) → 8 cm

1. Identify the Formulas

The area $A$ of a square can be calculated from its diagonal $d$ using the formula:

$A = \frac{d^2}{2}$

Conversely, the diagonal $d$ can be calculated from the area $A$ as:

$d = \sqrt{2A}$

2. Analyze the First Square

The problem states the diagonal of the first square is $4\sqrt{2}$ cm (interpreting the text "42 – √" as a common formatting error for $4\sqrt{2}$).

• Diagonal $d_1$: $4\sqrt{2}$ cm
• Area $A_1$: $\frac{(4\sqrt{2})^2}{2} = \frac{16 \times 2}{2} = 16$ cm$^2$

3. Analyze the Second Square

The area of the second square ($A_2$) is double the area of the first square ($A_1$).

• Area $A_2$: $2 \times A_1 = 2 \times 16 = 32$ cm$^2$

Now, we calculate the diagonal of the second square ($d_2$):

• Diagonal $d_2$: $\sqrt{2 \times A_2} = \sqrt{2 \times 32} = \sqrt{64} = 8$ cm

Alternative Method (Direct Ratio)

If the area of a square is doubled, the side length and the diagonal increase by a factor of $\sqrt{2}$.

• $d_2 = d_1 \times \sqrt{2}$
• $d_2 = (4\sqrt{2}) \times \sqrt{2}$
• $d_2 = 4 \times 2 = 8$ cm

Conclusion

The diagonal of the second square is 8 cm.