If the side of an square is a, then find the side of the regular octagon that can be inscribed in this square.

Options

A. A)`a(1+sqrt2)/sqrt2`
B. B)`a/3`
C. C)`a/2+sqrt2`
D. D)none of these

Answer

To find the side length of a regular octagon that can be inscribed in a square with side length \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Draw the Square**:
- Start by drawing a square and label its vertices as \( A, B, C, D \). Each side of the square has a length of \( a \).

2. **Inscribe the Octagon**:
- Now, inscribe a regular octagon within the square. Label the vertices of the octagon as \( P, Q, R, S, T, U, V, W \).

3. **Identify the Geometry**:
- The octagon will touch the midpoints of the sides of the square. Let’s denote the side length of the octagon as \( x \).

4. **Use Symmetry**:
- Due to the symmetry of the square and the regular octagon, the segments from the center of the square to the points where the octagon touches the square can be analyzed. The distance from the center of the square to the midpoint of a side of the square is \( \frac{a}{2} \).

5. **Analyze the Right Triangle**:
- Consider the right triangle formed by the center of the square, the midpoint of one side of the square, and one vertex of the octagon. The angles at the center are \( 45^\circ \) and \( 90^\circ \).

6. **Calculate the Lengths**:
- The lengths of the segments from the center to the vertex of the octagon can be expressed in terms of \( x \). The distance from the center to the vertex of the octagon is \( \frac{x}{\sqrt{2}} \) because of the \( 45^\circ \) angles.

7. **Set Up the Equation**:
- The total distance from the center to the midpoint of the square's side can be expressed as:
\[ \frac{a}{2} = x + \frac{x}{\sqrt{2}} \]

8. **Combine Terms**:
- Rewrite the equation:
\[ \frac{a}{2} = x \left(1 + \frac{1}{\sqrt{2}}\right) \]

9. **Solve for \( x \)**:
- Rearranging gives:
\[ x = \frac{a/2}{1 + \frac{1}{\sqrt{2}}} \]
- To simplify, multiply numerator and denominator by \( \sqrt{2} \):
\[ x = \frac{a \cdot \sqrt{2}/2}{\sqrt{2} + 1} \]
- This simplifies to:
\[ x = \frac{a \cdot \sqrt{2}}{2(\sqrt{2} + 1)} \]

10. **Final Simplification**:
- Rationalizing the denominator, we can express \( x \) in terms of \( a \):
\[ x = a \left(\sqrt{2} - 1\right) \]

### Final Answer:
The side length of the regular octagon inscribed in a square of side length \( a \) is:
\[ x = a \left(\sqrt{2} - 1\right) \]