Hi redcherry,

If you draw the right-triangle, with an arbitrary inscribed rectangle,

that has sides against the base and vertical side, settling in the

right-angled corner.

label the base of the triangle 6, height 8, hypotenuse 10.

The rectangle base is x and the rectangle height y.

Now, observe the smaller identical triangle at the bottom corner.

It has base (6-x) and height y.

Hence we can write an equation

Rectangle area is

in terms of x only, allowing us to differentiate the area and equate the derivative to zero to find the rectangle of maximum area

which equals zero when x=3