Lessons from a great geometry homework problem

My older son had a terrific homework problem in his enrichment math class. I wanted to walk through the problem again today so that my younger son could see it and also to highlight some of the lessons in the problem.

To start the project we revisted a fun geometry problem that will make a surprise appearance at the end of the homework problem:

Next I introduced the homework problem. My older son is familiar with this problem, but my younger son is seeing it for the first time. In this video my older son highlights the main ideas that we need to solve the problem (well . . . see the next video for the one we forgot!):

Here’s the one extra piece that we missed from the last video:

Next with the triangles labelled properly, we worked to see how we can use the Pythagorean theorem to help solve for the values of the two unknowns. I used this section of today’s project to give my younger son a little algebra practice:

Now comes the task of simplifying the two complicated equations. Hopefully that will help us make some progress towards solving them.

After the simplifying in the last video we are now ready to take a crack at solving for the radius of the smaller circles. Solving the equation involves solving a quadratic and that gave us a chance to talk about factoring.

Finally, we went back to the picture from the homework problem. We hadn’t solved for x in the project, but now we can use the pictures to help us find x’s value. We see an 8-15-17 triangle and also a 3-4-5 triangle. We also see the 5-5-8 triangle from the beginning of the project!

So, a fun project connecting a neat geometry problem that we’ve studied before with a new homework problem.

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