October 2024
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In the Math Corner of my October 2024 Diary I posted the following brainteaser.
Brainteaser: (Borrowed with thanks from Southall & Pantaloni's Geometry Snacks.)
Find the area of this triangle using a minimum of arithmetic and no trigonometry (sines, cosines, tangents, …) at all.
If you've spotted the key and want to win an A++ grade, work it out entirely in your head, no pen or paper.
• Solution.
The key is noticing that each of the numbers 13, 20, and 29 is the sum of two squares. In fact, each is the sum of 2² and another square: 3², 4², or 5².
Sum of two squares? Pythagoras! Yes: the three sides of that triangle are the hypoteneuses of three right-angled triangles with lesser sides (2, 3), (2, 4), and (2, 5).
At that point, for the A++ grade you must imagine this rectangle. If not up to that, you can easily sketch it.
It then very easily follows that the inscribed triangle has area 20 − (3 + 4 + 5), which is 8.