Proof and Problem Solving in Geometry

One of the fun things about home schooling for me is teaching subjects that I’ve never taught (or really thought about in years) for the first time. I’m in a section in our Geometry book about properties of triangles with my older son and it is an absolute blast work through with him.

It could easily be that I just wasn’t totally tuned in to the problem solving / proof process previously, but I feel as though I’m seeing my son pull together a lot of different ideas for the first time. It is so cool to see this process.

Last week we talked about this theorem – if a triangle has two equal medians, the triangle is isosceles. I just love watching the ideas from basic geometry come together here.

Tonight we looked at how you could find the radius of the circumscribed circle for an isosceles triangle. This is more problem solving-related than proof-related but I still really enjoyed watching the ideas come together. The extra connection with algebra here even brings in some math beyond geometry:

[a little post publication edit as I realized that you could use Patrick Honner’s Desmos program about circumscribed circles to explore the 2nd problem, too]

Patrick Honner has a nice Desmos program you can use to explore circumscribed circles. Setting the three points of the triangle in his program to be (0,0), (10,0), and (5,12) you can see (by clicking the “calculations” tab) that the radius of the circumscribed circle is indeed 169/24 = 7 1/24. Fun!

Patrick Honner’s Circumcircle program in Desmos

[end of post publication edit]

Before going through this Geometry book I’m not sure that I really understood why we were studying geometry. In fact, I’m sure that my answer wouldn’t have been that different from “that’s just what you study after algebra.” Now that we are in the middle of the book, though, I’m starting to get a better understanding of how studying geometry helps build your mathematical reasoning skills. It is so fun to watch those skills develop.

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