The midpoints of alternating sides of a regular octagon are joined to form a square. If each side of the octagon is a, find each side of the square.

Options

A. `(2+sqrt2)a/2`
B. `(4+sqrt2)a/2`
C. `2(1+sqrt2)a/sqrt2`
D. `(2+sqrt2)a`

Answer

To find the side of the square formed by joining the midpoints of alternating sides of a regular octagon, we can follow these steps:

### Step 1: Understand the Geometry of the Octagon
A regular octagon has 8 equal sides. Let's denote the length of each side as \( a \).

### Step 2: Identify the Midpoints
The midpoints of the alternating sides of the octagon will be selected. For an octagon with vertices labeled \( A_1, A_2, A_3, \ldots, A_8 \), we will take the midpoints of sides \( A_1A_2 \), \( A_3A_4 \), \( A_5A_6 \), and \( A_7A_8 \).

### Step 3: Calculate the Length of the Midpoints
Since the octagon is regular, the distance between two midpoints of alternating sides can be calculated. The distance between two midpoints (say \( M_1 \) and \( M_2 \)) can be determined using the properties of right triangles.

### Step 4: Form a Right Triangle
Consider one of the triangles formed by the midpoints. The angle between the sides of the octagon at the vertices is \( 45^\circ \) (since \( 360^\circ / 8 = 45^\circ \)). The line connecting the midpoints will form a right triangle with the sides of the octagon.

### Step 5: Use Pythagorean Theorem
Let’s denote the distance between two adjacent midpoints as \( PQ \). In triangle \( PQR \):
- \( PR \) (the distance between two midpoints) is \( \frac{a}{2} \).
- The height from the midpoint to the base is also \( \frac{a}{2} \).

Using the Pythagorean theorem:
\[ PQ^2 + PR^2 = QR^2 \]
Substituting the values:
\[ PQ^2 + \left(\frac{a}{2}\right)^2 = \left(\frac{a}{\sqrt{2}}\right)^2 \]
This simplifies to:
\[ PQ^2 + \frac{a^2}{4} = \frac{a^2}{2} \]
Thus,
\[ PQ^2 = \frac{a^2}{2} - \frac{a^2}{4} = \frac{a^2}{4} \]

### Step 6: Solve for \( PQ \)
Taking the square root gives us:
\[ PQ = \frac{a}{2\sqrt{2}} = \frac{a\sqrt{2}}{4} \]

### Step 7: Find the Side of the Square
The side of the square formed by the midpoints is equal to \( a + 2PQ \):
\[ \text{Side of the square} = a + 2\left(\frac{a\sqrt{2}}{4}\right) = a + \frac{a\sqrt{2}}{2} \]
Factoring out \( a \):
\[ \text{Side of the square} = a\left(1 + \frac{\sqrt{2}}{2}\right) \]

### Final Result
Thus, the side of the square formed by joining the midpoints of alternating sides of a regular octagon is:
\[ \text{Side of the square} = a\left(1 + \frac{\sqrt{2}}{2}\right) \]