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 Two triangles in a square (Posted on 2019-01-12)
Inside of a square ABCD, construct the largest equilateral triangle with one side on AB. Next, fit the largest equilateral triangle in the remaining space with one side on CD.

Find the ratio of the area of the smaller triangle to the largest.

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 Context for the problem Comment 2 of 2 |
https://blogs.scientificamerican.com/roots-of-unity/how-to-celebrate-thirdsday/

I found out about Thirdsday the evening before.  I knew I had to come up with something for each of my classes the following day.  For my geometry class I decided to give them each a square of paper and show them a fold that results in a 30 degree angle (30 being a third of 90)  When they finished I challenged them to fold up their piece of paper to show an equilateral triangle.

About a third of the class came up with the larger triangle described and another third the smaller (the rest couldn't figure it out.)  I immediately looked and guessed the ratio was 1/3 but I would have to take some time to figure it out.

At the end of the day, one enterprising student came to me and she had solved it.

I removed the origami component to share it here on Perplexus.  I also removed the reference to the date, as it was too late to have it published on the day.

I made a sheet of directions so I could use this problem in the future.  I posted a picture to Brilliant and made it into a problem there.

https://brilliant.org/problems/an-origami-problem-for-thirdsday/?ref_id=1558682

 Posted by Jer on 2019-01-12 12:29:17
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