Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths "4 cm and 5 cm" if two sides of the rectangle lie along the legs.

If the area of the rectangle = xy  we first have to find a relation between x and y.

If the angle opposite the side of length 5 is A then we have
tan A = 5/4

so we have the relation
5/4 = x / (4 - y)
giving
x = 5(4 - y) / 4

Substituting in  the formula for the area ( A = xy), we have
 A  = 5y(4 - y) / 4
A = 5y - 1.25y^2  
Differentiating:-
dA / dy  =  5 - 2.5y   = 0  for maximum area
y  = 2 

and x = 5(4 - 2) / 4 =  2.5  

So area of largest rectangle = 2 * 2.5  = 5 cm^2