Applets vs Algebra: Triangle Ratios

I’ve been doing a lot of work on a Master’s assignment over the past month which has prevented me from posting, but I stumbled upon a problem the other day which I thought I’d quickly share.

One equilateral triangle (DEF) is inscribed into another equilateral triangle (ABC). You can change the size of DEF by moving the slider (Click the picture below to go the applet).

What is the relationship between the ratio of the lengths of each triangle and the ratio of the areas? This could be calculated algebraically or could be investigated in geogebra depending on the ability of the students. The relationship ends up being quadratic. Can you figure out why this is the case? Whilst the applet does bring the problem to “life” and makes it easier to investigate, it does inhibit the reasoning behind why the connection is quadratic which an algebraic approach would easily confirm. There was a big debate quite recently on whether we need really need algebra anymore. It is the process of algebraic derivation which allows us the glimpse into the inner workings of this problem and understand the formulae.


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