The ratio of the area of a square to that of the square drawn on its diagonal is :

Options

A. `1:1`
B. `1:2`
C. `1:3`
D. `1:4`

Answer

To find the ratio of the area of a square to that of the square drawn on its diagonal, let's follow these steps:

### Step 1: Define the side of the square
Let the side length of the square be \( a \).

### Step 2: Calculate the area of the square
The area \( A \) of the square is given by the formula:
\[ A = \text{side}^2 = a^2 \]

### Step 3: Calculate the length of the diagonal
Using the Pythagorean theorem, the length of the diagonal \( d \) of the square can be calculated as:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 4: Calculate the area of the square drawn on the diagonal
The area of the square drawn on the diagonal is given by:
\[ \text{Area of square on diagonal} = d^2 = (a\sqrt{2})^2 = 2a^2 \]

### Step 5: Calculate the ratio of the areas
Now, we need to find the ratio of the area of the original square to the area of the square on the diagonal:
\[ \text{Ratio} = \frac{\text{Area of square}}{\text{Area of square on diagonal}} = \frac{a^2}{2a^2} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the area of the square to that of the square drawn on its diagonal is:
\[ 1 : 2 \]