The rectangle is inscribed in a right angled triangle.

$\"\"$

$\"\"$ and $\"\"$ are the side of the rectangle.

The sides of the triangle are $\"\"$ and $\"\"$ .

Two points form of a line equation is $\"\"$ .

The two points are $\"\"$ and $\"\"$ is

$\"\"$

The area of the rectangle is $\"\"$ .

$\"\"$

The area of the rectangle is maximum when its first derivative becomes zero.

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

Equate $\"\"$ to zero.

$\"\"$

The length of the rectangle is $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Therefore the area of the largest rectangle that is inscribed in a right angled triangle is $\"\"$ .

The area of the largest rectangle that is inscribed in a right angled triangle is $\"\"$ .