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

Options

A. A)`(sqrt(2+sqrt2))a`
B. B)`(1+sqrt2)a`
C. C)`2(1+sqrt2)a`
D. D)`(sqrt4+root(2)(2))a`

Answer

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

### Step 1: Understand the properties of a regular octagon
A regular octagon has eight equal sides and eight equal angles. The interior angle of a regular octagon is \( 135^\circ \).

### Step 2: Identify the shortest diagonal
The shortest diagonal in a regular octagon connects two non-adjacent vertices that are closest to each other. For example, if we label the vertices of the octagon as \( A_1, A_2, A_3, A_4, A_5, A_6, A_7, A_8 \), the diagonal \( A_1A_3 \) would be the shortest diagonal.

### Step 3: Use the cosine rule
To find the length of the diagonal \( A_1A_3 \), we can apply the cosine rule. The cosine rule states that for any triangle with sides \( a, b, c \) opposite to angles \( A, B, C \), respectively:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
In our case, we can consider the triangle formed by the vertices \( A_1, A_2, A_3 \). Here, \( A_1A_2 = a \), \( A_2A_3 = a \), and the angle \( A_2 \) is \( 135^\circ \).

### Step 4: Apply the cosine rule
Let \( k \) be the length of the diagonal \( A_1A_3 \). According to the cosine rule:
\[ k^2 = a^2 + a^2 - 2 \cdot a \cdot a \cdot \cos(135^\circ) \]
Since \( \cos(135^\circ) = -\frac{1}{\sqrt{2}} \), we can substitute this value into the equation:
\[ k^2 = 2a^2 - 2a^2 \left(-\frac{1}{\sqrt{2}}\right) \]
\[ k^2 = 2a^2 + \frac{2a^2}{\sqrt{2}} \]

### Step 5: Simplify the expression
Now, we can simplify the expression:
\[ k^2 = 2a^2 + \sqrt{2}a^2 \]
\[ k^2 = a^2(2 + \sqrt{2}) \]

### Step 6: Take the square root
To find \( k \), we take the square root of both sides:
\[ k = a\sqrt{2 + \sqrt{2}} \]

### Conclusion
Thus, the length of the shortest diagonal of the octagon is:
\[ \boxed{a\sqrt{2 + \sqrt{2}}} \]