Isosceles Right Triangle Dissection

In my opinion, some of the best ideas for a student Exploration (IA) come from a challenging problem that requires creative application of mathematical concepts and skills covered in the HL and/or SL course.  While solving such a problem, if a student (with some encouragement and guidance from his/her teacher) can think of their own way to extend the problem, or consider a related problem, then they may have come up with a good idea for an Exploration.  For a student to formulate their own questions that can lead to a rich problem with an interesting, or surprising, result can be a very rewarding experience for students - and can lead to a high-quality Exploration.  Coming up with appropriate questions is not easy for students - and teachers should be prepared to offer timely suggestions (without doing all the thinking for the student) to help students develop their own ideas.

This particular Exploration idea starts with Problem #101 on the HL Problem of the Day page.  Although I originally made it an HL P.o.t.D., I do think that an above average SL student could turn this problem into a succesful Exploration.  Below is the original problem.

The full solution is given on the 2-page document 101_HL_Problem_10_Jan_2018.  The first key result in the solution comes from developing expressions for the area of triangles ABC and ADE - and showing that \(xy = 25\sqrt 2 \).  Letting \({\textrm{DE}} = z\), applying the cosine rule to triangle ADE and substituting for \(y\) in terms of \(x\) eventually produces the equation \({z^2} = {x^2} + \frac{{1250}}{{{x^2}}} - 50\).  Applying a bit of calculus gives the result that for triangle ADE to have half the area of triangle ABC, and DE to be as short as possible, then \(x\) and \(y\) are both equal to \(5\,\sqrt[4]{2} \approx 5.94604\) and DE\( \approx 4.5509\).

This problem - on its own - is not enough for a proper Exploration.  However, the questions that you would like a student to ask after solving this dissection problem are:  This is the result for when angle BAC is 45 degrees ... what happens for other angles?  What is the solution for other 'nice' measures of angle BAC, e.g. when it's 30 degrees, or 60 degrees?  Is there a pattern?  If angle BAC is represented by a variable, is there a general solution?  Will the solution for any measure of angle BAC always result in triangle ADE being isosceles, that is \(x = y?\)

This is not a well-known problem.  As far as I can tell, it's not really "known" at all - and this is advantageous for the student; although, they may not necessarily agree (initially, at least).  But the fact that an internet search does not result in any immediate answers to the questions posed above forces the student to work it out for themselves, come up with their own questions & ideas, and to express all of this in their own words.

I would encourage a student to explore their extension questions for this dissection problem using more robust technology than just their GDC.  I created the Geogebra applet below to explore further aspects of this dissection problem.  I would not just give this applet to a student for them to use in their Exploration.  I might briefly show it to them and then give them encouragement and advice for them to create their own dynamic applet (personal engagement).  I always spend some class time (mostly in the 1st year) showing students a few tech tools that I consider to be very helpful (but not required) when working on their Explorations.  These brief tech tutorials always include MathType (equation editor) and Geogebra.  I would not worry if you, the teacher, are not familiar with Geogebra - and you'd like students to consider using it - because there is a great deal of support (tutorials, manual, forum, blog, etc) on the Geogebra website.

In the applet, angle BAC (labelled \({\theta }\)), can be changed by either using the slider or by entering a value into the input box to the right of the slider.  The slider is nice for seeing the triangle change shape, but being able to enter a value for \(\theta \) is helpful when wishing to investigate what happens for a specific value.  The same can be done for the lengths AD and AE - change their values using a slider or enter a specific value into the appropriate input box.

A student could generate results for different measures of angle BAC and plot these to explore if there is any kind of pattern or general result.

An analytic approach where the variable \(\theta \) represents angle BAC and, again, substituting for \(y\) in terms of \(x\) produces the following equation that relates the minimum length for DE and \(x\) and \(\theta \) :  DE \( = \sqrt {{x^2} + \frac{{625}}{{{x^2}}}{{\sec }^2}\theta - 50} \)

This equation contains three unknowns so it does not have a representation in a 2D coordinate plane.  But it's possible to produce a 2D graph for each specific value of \(\theta \).  With software such as Geogebra, Desmos, TI-Nspire and others, it's quite easy to show the 'family' of solution curves for DE by making \(\theta \) a constant and changing its value using a slider, or entering a value into an input box.  I created the Geogebra applet below to do this.  Do you think there is a general result?

Please share any feedback on this Exploration idea in the Comments at the bottom of the page.  Thank you.