09. MENSURATION
Explanation:
4 : 1
Let the two squares be ABCD and PQRS. Further, the diagonal of square PQRS is twice the diagonal of square ABCD.

PR = 2AC
Now, area of the square \(=\frac{(\text{diagonal})^{2}}{2}\)
Area of PQRS \(=\frac{(\text{PR})^{2}}{2}\)
Similarly, area of ABCD \(=\frac{(\text{AC})^{2}}{2}\)
According to the question:
If AC = x units, then, PR = 2x units
Therefore, \(\frac{\text{Area of PQRS}}{\text{Area of ABCD}}=\frac{(\text{PR})^{2}\times2}{2\times(\text{AC})^{2}}\)
\(=\frac{(\text{PR})^{2}}{(\text{AC})^{2}}=\frac{(2\text{x})^{2}}{(1\text{x})^{2}}=\frac{4}{1}\)
\(=4:1\)
Thus, the ratio of the areas of squares PQRS and ABCD = 4 : 1