Exploring Area and Perimeter

At the RMS Spring Symposium, Dr. Christine Suurtamm used the following scenario as a springboard to talk about Math content knowledge and pedagogical content knowledge (i.e., what is important mathematically and how to help a student learn about it).

There was lots of rich discussion.  It made me think about taking three sides of a polygon and moving the middle side out.  Here the right side of the square has been moved out to make a rectangle with larger area and larger perimeter.  I called this "extrusion" but am not sure that is mathematically correct and became uninterested in whether the new side was parallel to the old one.

For a simple quadrilateral, it makes sense to me that as you move the middle side out, the area gets bigger.

To keep the notation friendlier, let's call newA "C" and newB "D".

The perimeter looks like it is getting bigger (i.e., that AC + CD + DB is bigger than AB) but I wondered if the decrease in CD might be bigger than the increase along the ray formed by the containing sides (AC + DB).

My buddy Greg used a triangle inequality argument to convince me that the perimeter gets bigger. The triangle inequality is a fancy way of saying that the shortest distance between two points is a line. Here the distance from A to D is shortest along the line, going via C is longer.

AC + CD > AD

Similarly, looking at triangle ABD,

AD + DB > AB

Putting it together,

AC + CD + DB > AD + DB > AB

So the new perimeter is bigger than the old one.

I started to feel smug and accomplished and then I thought about a counter-example,

As a teacher, where would you go with this?