A square whose side is 2 metres has its corners cut away so as to form an octagon with all sides equal.
Then, the length of each side of the octagon in metres is :

Options

A. `(sqrt(2))/(sqrt(2)+1)`
B. `(2)/(sqrt(2)+1)`
C. `(2)/(sqrt(2)-1)`
D. `(sqrt(2))/(sqrt(2)-1)`

Answer

To find the length of each side of the octagon formed by cutting the corners of a square with a side length of 2 meters, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a square with each side measuring 2 meters. When we cut off the corners of the square, we form an octagon. We need to find the length of each side of this octagon.

2. **Define Variables**:
Let the length of each side of the octagon be \( x \) meters.

3. **Identify Relationships**:
When we cut off the corners of the square, we create right triangles at each corner. The lengths of the legs of these triangles will be equal due to symmetry. Let’s denote the length of each leg of the triangle as \( a \).

4. **Use Pythagorean Theorem**:
In one of the right triangles formed, we can apply the Pythagorean theorem:
\[ LE^2 = AL^2 + AE^2 \]
Since \( AL = AE = a \), we can rewrite this as:
\[ x^2 = a^2 + a^2 = 2a^2 \]
Thus,
\[ x = a\sqrt{2} \]

5. **Express the Side of the Square**:
The side of the square can be expressed in terms of \( a \) and \( x \):
The total length of one side of the square (2 meters) can be represented as:
\[ 2 = AL + GH + HC = a + x + a = 2a + x \]
Substituting \( x = a\sqrt{2} \) into the equation gives:
\[ 2 = 2a + a\sqrt{2} \]

6. **Rearranging the Equation**:
Rearranging the equation:
\[ 2 = a(2 + \sqrt{2}) \]
Solving for \( a \):
\[ a = \frac{2}{2 + \sqrt{2}} \]

7. **Substituting Back to Find \( x \)**:
Now substitute \( a \) back into the equation for \( x \):
\[ x = a\sqrt{2} = \frac{2\sqrt{2}}{2 + \sqrt{2}} \]

8. **Rationalizing the Denominator**:
To simplify \( x \):
\[ x = \frac{2\sqrt{2}(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})} = \frac{2\sqrt{2}(2 - \sqrt{2})}{4 - 2} = \frac{2\sqrt{2}(2 - \sqrt{2})}{2} = \sqrt{2}(2 - \sqrt{2}) \]

9. **Final Calculation**:
Thus, the length of each side of the octagon is:
\[ x = 2 - \sqrt{2} \text{ meters} \]

### Final Answer:
The length of each side of the octagon is \( 2 - \sqrt{2} \) meters.