Friday, January 15, 2016

Sometimes its Harder the other way Part 2

I spent some time thinking about what initially looked like a very simple triangle congruence problem last night which I've outlined below. Given a perpendicular angle bisector of a triangle, show that it intersects the bottom at its median. Turn the problem on its head, (Given a perpendicular bisector of a triangle that intersects the median prove its perpendicular) and it suddenly become quite a bit trickier since you lose the obvious way to show the triangles are congruent.

What I like here is that its very similar structurally to this problem:
Both problems have the same asymmetric complexity. Both are solvable by essentially reflecting above or below the original figure.  Even better, this version is not as hard the previous one which would make  for a very nice progression if presented in sequence.

(Apology: I tried doing the whole explanation in Geogebra which doesn't scale as well as I intended. I recommend zooming in to read the figures)


  1. It is worth thinking about what SSA actually does tell you.

    While SSA usually isn't enough information to specify a single congruence class of triangles, it does get you down to only two choices that have some relationships to each other. In this problem, those triangles have angles that are supplementary, so you can work with that and find they have to be right angles.

    One way to build intuition in the kids is to do constructions. Given data (SSS, SAS, ASA, SSA) can they construct triangles that fit the data. Do they always get the "same" triangle or not? Where did they feel they had a choice of what to do?

    You can also throw in impossible data sets, e.g., 3 lengths that don't satisfy the triangle inequalities, thus don't correspond to a triangle. For which data sets (SSS, SAS, ASA, SSA) do you always get a triangle (or more) that fit?

    Not sure if this type of activity would fit your math club or is better in a standard class.


    1. The kids I see will not hit congruent triangles/geometry until 7th or 8th grade. So this is mostly a mental exercise for now. In the meantime there is plenty of fun stuff within their grasp. I do anticipate graduating to middle school math clubs at some point though