please help on the applied maximum problem?

Question

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3cm and 4cm if two sides of the rectangle lie along the legs.

Answer 1

The triangle is a right triangle of sides 3,4,5.
The maximum rectangle occurs when a perpendicular is raised from the midpoint of the short side to the hypotenuse (side that equals 5) and then a perpendicular from point to the side that is of length 4. The dimensions of this rectangle are 2* 1.5 = 3 square units.