What is the greatest area of a rectangle inscribed in a given right-angled triangle?

Consider this situation, where \(C\) is a vertex of both the rectangle and the triangle.

The 2 vertices adiacent to C are on A C and C B and the fourth vertex is on A B.
The length of the side parallel to C A is x.

If the horizontal length of the rectangle is labelled \(x\), what is the vertical length of the rectangle in terms of \(x\) and side lengths \(a\) and \(b\)?

Consider this situation, where one edge of the rectangle lies on the line \(AB\).

The third vertex is on C B and the fourth is on C A.
  • How does your approach to each situation compare?

  • How does the area of the rectangle identified in each situation compare?