Dave Radcliffe’s Geometry problem

Saw a neat tweet from Dave Radciffe earlier in the week:

I was reminded of this problem again this morning when I asked my older son what he wanted to talk about for our Family Math project today. The topic he chose was similarity and congruence. I was a little surprised since it has been a while since we’ve talked about any geometry, but Dave’s problem fit the bill.

Although this problem is a little to difficult for my younger son to understand, I thought it would be fun to talk through it with both kids. He did struggle with a few of the ideas, but I think that overall it was an interesting problem for him. For my older son, this was a nice geometry review.

We started by talking through the problem and finding one easy idea:

Next we returned to the more general case to explore the ideas of similarity and congruence that help solve this problem:

For the next part of the project we tried to apply the ideas from the last part of the talk to our problem. This piece was difficult for my younger son, but we went slowly. It was nice to see him begin to understand some of the algebraic expressions.

Finally, we wrapped up by finishing the calculations and found that the area the shaded region in the problem was 4 \pi no matter where the segment of height 4 is placed!

So, a fun project and a excellent bit of luck that Dave had posted this problem earlier in the week. I still don’t know what made my son think of today’s topic, but it was a good geometry review for him and a nice algebra / geometry introduction for my younger son.

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