If each side of the octagon be a, then find the longest diagonal of the octagon.

Options

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

Answer

To find the longest diagonal of a regular octagon where each side is of length \( a \), we can follow these steps:

### Step 1: Understand the Structure of the Octagon
A regular octagon has 8 sides, and we can label the vertices as \( A, B, C, D, E, F, G, H \). The longest diagonal in the octagon will connect two vertices that are not adjacent and are as far apart as possible.

### Step 2: Identify the Longest Diagonal
The longest diagonal will be from vertex \( A \) to vertex \( C \) (or any two vertices that are opposite each other). We will denote this diagonal as \( AC \).

### Step 3: Divide the Octagon into Triangles
To find the length of diagonal \( AC \), we can use the properties of triangles formed by the sides of the octagon. We can drop perpendiculars from points \( B \) and \( H \) to line \( AC \).

### Step 4: Use the Pythagorean Theorem
Let \( B \) and \( H \) be the midpoints of the sides adjacent to \( A \) and \( C \). The distance \( AB \) and \( AH \) can be calculated using the Pythagorean theorem.

1. The distance from the center of the octagon to any vertex (radius \( R \)) can be calculated as:
\[ R = \frac{a}{2 \sin(\frac{\pi}{8})} \]

2. The length of diagonal \( AC \) can be calculated using:
\[ AC = 2R \cos(\frac{\pi}{8}) \]

### Step 5: Substitute the Radius
Substituting the value of \( R \) into the equation for \( AC \):
\[ AC = 2 \left( \frac{a}{2 \sin(\frac{\pi}{8})} \right) \cos(\frac{\pi}{8}) = \frac{a \cos(\frac{\pi}{8})}{\sin(\frac{\pi}{8})} \]

### Step 6: Simplify the Expression
Using the identity \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \):
\[ AC = a \cot(\frac{\pi}{8}) \]

### Step 7: Final Expression
The length of the longest diagonal \( AC \) in terms of \( a \) is:
\[ AC = a \sqrt{4 + 2\sqrt{2}} \]

### Conclusion
The longest diagonal of the octagon is given by the expression:
\[ AC = a \sqrt{4 + 2\sqrt{2}} \]