The length of the side of a square is 14 cm. Find out the ratio of the radii of the inscribed and circumscribed circle of the square.

Options

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

Answer

To solve the problem of finding the ratio of the radii of the inscribed and circumscribed circles of a square with a side length of 14 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the side length of the square**:
- The side length \( a \) of the square is given as 14 cm.

2. **Calculate the radius of the inscribed circle (r)**:
- The radius of the inscribed circle (also known as the incircle) of a square is given by the formula:
\[ r = \frac{a}{2} \]
- Substituting the value of \( a \):
\[ r = \frac{14}{2} = 7 \text{ cm} \]

3. **Calculate the radius of the circumscribed circle (R)**:
- The radius of the circumscribed circle (also known as the circumcircle) of a square is given by the formula:
\[ R = \frac{a}{\sqrt{2}} \]
- Substituting the value of \( a \):
\[ R = \frac{14}{\sqrt{2}} = 7\sqrt{2} \text{ cm} \]

4. **Find the ratio of the radii**:
- The ratio of the radius of the inscribed circle to the radius of the circumscribed circle is:
\[ \text{Ratio} = \frac{r}{R} = \frac{7}{7\sqrt{2}} \]
- Simplifying this ratio:
\[ \text{Ratio} = \frac{1}{\sqrt{2}} \]

5. **Express the ratio in a standard form**:
- The ratio can be expressed as:
\[ 1 : \sqrt{2} \]

### Final Answer:
The ratio of the radii of the inscribed and circumscribed circles of the square is \( 1 : \sqrt{2} \).

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