# How do you find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 8 cm if two sides of the rectangle lie along the legs?

##### 1 Answer

Jul 23, 2017

Its area is

#### Explanation:

The largest possible rectangle must have a vertex that touches the hypotenuse of the triangle at one point.

Let us put the right angle of the triangle at the point

We can represent points on the hypotenuse parametrically as:

#(8t, 3(1-t))#

where

Then the area of the rectangle with vertices:

#(0, 0)# ,#(8t, 0)# ,#(8t, 3(1-t))# ,#(0, 3(1-t))# is:

#f(t) = 8t*3(1-t) = 24t(1-t) = 24(t-t^2) = 24(1/4-(t-1/2)^2)#

This takes its maximum value when