Question

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

a.8cm
b.$8\sqrt{2}$ cm
c.16cm
d.$4\sqrt{2}$ cm

Answer 1

Hint: We know that the diagonal of a square is $\sqrt{2}$ times its side, or $diagonal=\sqrt{2}\times sides$. From this we will find the area of the first square and then we will find the area of the second square by doubling it and from that area we will find the sides of the second square and hence the diagonal of the second square.

Complete step-by-step answer:

It is given in the question that the diagonal of the first square is $4\sqrt{2}$ cm, and we have to find the diagonal of the second square. We know that the diagonal of a square is $\sqrt{2}$ times the sides of any square, or $diagonal=\sqrt{2}\times sides$. Since the diagonal of square is $4\sqrt{2}$cm, so from this, we get side of square = $\frac{diagonal}{\sqrt{2}}$ = $\frac{4\sqrt{2}}{\sqrt{2}}$ cm = 4 cm. Therefore, the side of the square = 4cm.

Now, we know that area of square = ${{\left( side \right)}^{2}}$. Therefore, area of first square = ${{\left( 4 \right)}^{2}}$ = $16c{{m}^{2}}$. Now, the area of the second square is given to be double of the first. Therefore, the area of the second square = 16cm × 2cm = $32c{{m}^{2}}$.
We know that the area of the square is the square of its sides. Therefore the side of the second square = $\sqrt{area}$ = $\sqrt{32}$ =$4\sqrt{2}$ cm. From equation $diagonal=\sqrt{2}\times sides$, we get its diagonal = $\sqrt{2}\times 4\sqrt{2}=4\times 2=8cm$.

Therefore, the diagonal of the second square will be 8cm. Hence option a) is correct.

Note: It is important to know the relation between the sides of the square and its diagonal, which is $diagonal=\sqrt{2}\times sides$. The student can make a mistake by taking the multiplication factor of 2 instead of factor $\sqrt{2}$.