Find the ratio of the areas of squares circumscribed about and inscribed in the same circle.

Options

A. `1:3`
B. `2:1`
C. `sqrt(2) :1`
D. `1:sqrt(2)`

Answer

To find the ratio of the areas of squares circumscribed about and inscribed in the same circle, we can follow these steps:

### Step 1: Define the radius of the circle
Let the radius of the circle be denoted as \( r \).

### Step 2: Determine the side length of the circumscribed square
A square that is circumscribed about the circle will have its sides equal to the diameter of the circle. The diameter \( D \) is given by:
\[ D = 2r \]
Thus, the side length \( S_{circumscribed} \) of the circumscribed square is:
\[ S_{circumscribed} = 2r \]

### Step 3: Calculate the area of the circumscribed square
The area \( A_{circumscribed} \) of the circumscribed square can be calculated using the formula for the area of a square \( A = \text{side}^2 \):
\[ A_{circumscribed} = (S_{circumscribed})^2 = (2r)^2 = 4r^2 \]

### Step 4: Determine the side length of the inscribed square
A square that is inscribed in the circle will have its diagonal equal to the diameter of the circle. The diagonal \( D \) of the inscribed square is also \( 2r \). If \( S_{inscribed} \) is the side length of the inscribed square, we can relate the diagonal to the side length using the formula:
\[ D = S_{inscribed} \sqrt{2} \]
Setting the diagonal equal to the diameter:
\[ 2r = S_{inscribed} \sqrt{2} \]
Solving for \( S_{inscribed} \):
\[ S_{inscribed} = \frac{2r}{\sqrt{2}} = r\sqrt{2} \]

### Step 5: Calculate the area of the inscribed square
The area \( A_{inscribed} \) of the inscribed square is:
\[ A_{inscribed} = (S_{inscribed})^2 = (r\sqrt{2})^2 = 2r^2 \]

### Step 6: Find the ratio of the areas
Now, we can find the ratio of the areas of the circumscribed square to the inscribed square:
\[ \text{Ratio} = \frac{A_{circumscribed}}{A_{inscribed}} = \frac{4r^2}{2r^2} = \frac{4}{2} = 2 \]

Thus, the ratio of the areas of the squares circumscribed about and inscribed in the same circle is:
\[ \text{Ratio} = 2:1 \]