If a rectangle is inscribed in a equilateral triangle of side `2sqrt2` then side of maximum area of rectangle is

Answer

To find the side of the rectangle of maximum area inscribed in an equilateral triangle with side length \(2\sqrt{2}\), we can follow these steps:

### Step 1: Understand the Geometry
We have an equilateral triangle \(ABC\) with side length \(2\sqrt{2}\). We inscribe a rectangle \(PQRS\) within this triangle. Let the base of the rectangle \(PQ\) lie on side \(AB\) and the vertices \(R\) and \(S\) touch sides \(AC\) and \(BC\) respectively.

### Step 2: Define Variables
Let the length of the rectangle be \(SR = PQ = 2x\) (since \(P\) and \(Q\) are at the same horizontal level) and the height \(QR\) be denoted as \(h\). The height of the triangle can be determined using the formula for the height of an equilateral triangle:
\[ h_{triangle} = \frac{\sqrt{3}}{2} \times \text{side} \]
Substituting the side length:
\[ h_{triangle} = \frac{\sqrt{3}}{2} \times 2\sqrt{2} = \sqrt{6} \]

### Step 3: Relate Height to Rectangle Dimensions
The height \(h\) of the rectangle can be expressed in terms of \(x\). The height of the triangle above the rectangle is:
\[ h = \sqrt{6} - h_{rectangle} \]
Using the properties of the triangle, we can find the height of the rectangle as:
\[ h_{rectangle} = \sqrt{3} \cdot ( \text{base} ) = \sqrt{3} \cdot ( \sqrt{2} - x ) \]
Thus, we have:
\[ h = \sqrt{6} - \sqrt{3}(\sqrt{2} - x) \]

### Step 4: Area of the Rectangle
The area \(A\) of the rectangle can be expressed as:
\[ A = \text{length} \times \text{height} = 2x \cdot h \]
Substituting for \(h\):
\[ A = 2x \left(\sqrt{6} - \sqrt{3}(\sqrt{2} - x)\right) \]

### Step 5: Simplify the Area Expression
Expanding this:
\[ A = 2x\sqrt{6} - 2x\sqrt{3}(\sqrt{2} - x) \]
\[ A = 2x\sqrt{6} - 2x\sqrt{6} + 2x^2\sqrt{3} \]
\[ A = 2x^2\sqrt{3} \]

### Step 6: Maximize the Area
To find the maximum area, we differentiate \(A\) with respect to \(x\):
\[ \frac{dA}{dx} = 4x\sqrt{3} \]
Setting the derivative equal to zero to find critical points:
\[ 4x\sqrt{3} = 0 \implies x = 0 \]
However, we also need to check the endpoints of the interval for \(x\) (from \(0\) to \(\sqrt{2}\)).

### Step 7: Calculate Maximum Area
To find the maximum area, we can also consider the second derivative test or check values of \(x\) at critical points. We find:
\[ x = 1 \quad \text{(from previous calculations)} \]
Thus, the side of the rectangle is:
\[ SR = 2x = 2 \cdot 1 = 2 \]

### Final Answer
The side of the rectangle of maximum area inscribed in the equilateral triangle is \(2\).