Back-to-School Geometry - Rectangles, Squares and Deeper Challenges

The connections between geometry and other rich areas of mathematics are boundless. Here is a fairly straightforward set of problems that can be explored as far as your eye and mind can see. On the surface, we have three rectangles each of which has a half-diagonal of length 6. Students can be asked to find the area of each without using any trigonometry. On a deeper level, one can ask students to explain or prove why the square has the greatest area for a given
diagonal length. This is straightforward using the well-known trig formula for area of a triangle, K = (1/2)absin C, however the challenge here is to use non-trig methods (although the student can use special right triangles) to compute the areas and demonstrate the maximum. The maximum piece is more sophisticated and the idea of bringing this in before precalculus and calculus has many benefits.

The instructor might begin by asking students to draw any rectangle with a diagonal of 12. How many such rectangles could there be? Which one would appear to have the greatest area? Ok, now let's explore a few special cases.

This problem allows the creative student to devise a visual way of explaining the maximum. It also allows the instructor to bring in the Arithmetic Mean-Geometric Mean Inequality for enrichment. So many methods and approaches are possible...

Just Another Square Problem? A Means to an End...

Before announcing the thousands (or less!) winners of the Name That Mathematician Challenge, I came across a problem about dissecting a square ABCD with lines PQ and RS which are parallel to the sides of the square. (see diagram).

Naturally, I decided to make it into a deeper investigation. Students and/or readers will be asked to find the maximum value of the product of the areas of either pair of non-adjacent rectangles formed. There are many approaches here, one of which uses the famous Arithmetic Mean-Geometric Inequality. As usual you will work from the particular to the general, beginning with a specific value for the sides of ABCD.

STUDENT/READER INVESTIGATION - PART I
The given conditions about the diagram are given above.

For Part I, we will assume each side of the square has length 4.

(1) (Particular) If AP = 3 and RC = 2, determine the product of the areas of APTS and RTQC. Do the same for the other pair of non-adjacent rectangles formed. Do you believe this product is the maximum possible as we vary the positions of segments PQ and RS?

(2) (General) Show that the product of the areas of either pair of non-adjacent rectangles formed is less than or equal to 16. For example the product of the areas of APTS and RTQC is ≤ 16.

Notes:
(1) Do you think many students would guess what the configuration would be for the maximum product to occur? Is proving the conjecture much more difficult?
(2) The challenge here is to find an effective use of variables to denote the segments. There are many possibilities, some much more efficient than others.
(3) I will add additional parts to this challenge after receiving comments on Part I. How would you generalize this result further? More interestingly, there is a way to prove Part I using the AM-GM Inequality?