If the diagonal of a square is made 1.5 times, then the ratio of the areas of two squares is

### Concept & Formula The area of a square can be calculated directly from its diagonal using the formula: $$ \text{Area} = \frac{d^2}{2} $$ where $d$ is the length of the diagonal. ### Step-by-Step Solution 1. Let the diagonal of the original square be $d$. 2. The area of the original square is $A_1 = \frac{d^2}{2}$. 3. The diagonal of the new square is made $1.5$ times, so it is $1.5d = \frac{3}{2}d$. 4. The area of the new square is $A_2 = \frac{ (3d/2)^2 }{ 2 } = \frac{ \frac{9d^2}{4} }{ 2 } = \frac{9d^2}{8}$. 5. The ratio of the areas of the original square to the new square is $A_1 : A_2 = \frac{d^2}{2} : \frac{9d^2}{8}$. 6. Simplify the ratio by multiplying both sides by $\frac{8}{d^2}$: $4 : 9$. ### Exam Strategy & Shortcut Since Area is proportional to the square of any linear dimension (like the diagonal), if the diagonal is multiplied by a factor $k$, the area is multiplied by $k^2$. Here, $k = 1.5 = \frac{3}{2}$. Therefore, the area factor is $(\frac{3}{2})^2 = \frac{9}{4}$. The ratio of original area to new area is $1 : \frac{9}{4}$, which simplifies to $4 : 9$. ### Common Pitfall Confusing the order of the ratio. The question asks for the ratio of the areas of the two squares, implying original to new. The inverse ratio $9 : 4$ is not among the options, which helps prevent this error. ### Final Answer Therefore, the correct answer is **4 : 9**.