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