Optimization with maxima and minima

[electritron]

A rectangle is to be inscribed in a right triangle having sides 6 inches, 8 inches, and 10 inches. Determine the dimensions of the rectangle with greatest area.

I recently tried doing it and the answer was found by finding the slope and then using the first and second derivatives of the area.
Now I would like to know how to solve this by using proportions.
So if there are any suggestions whatsoever, Thanks.

 

[epenguin]

I confess I was not sure what is meant by a rectangle inscribed in...
Didi it mean a rectangle with base on the hypotenuse, or one with the same right angle as the right triangle?

After working out an answer for one I realized the answer would be the same for the other!

What is your answer?

You are given a classical 3, 4, 5 right triangle - but does your result really depend on that? As a fraction of the total area of the triangle can you state maximum area of inscribed rectangle in the most general fashion, i.e. for any right triangle?

I cannot think of how to do it by proportions, but if you consider the area of strip added on one side and subtracted on the other side as you move a corner of the rectangle I think you can get the result without formal differential calculus, indeed without having to consider the strips to be narrow.