Each month in 2020 I have featured a different type of logic puzzle. This month it’s…
The Nearly Ubiquitous 3-4-5
“Uncle Bob” Mead
In this last installment of my 2020 survey of various “flavors” of logic, I return to my first high-stakes encounter, that being with high school geometry. Proof equals panic? I admit I struggled at first with the notions of logic and proof. I credit my teacher Gloria Schwartz with a gem of an idea. She introduced us to some of the other flavors: “The engineer, the conductor, and the fireman were named Smith, Brown, and Jones. Smith lived in Boston and the fireman had red hair, and ….” After that exposure my proofs were more successful, more airtight. And so …
By 3-4-5 I mean, of course, the right triangle with those side measures. I can make the “nearly ubiquitous” case with one exhibit. The 3-4-5 is used as an example when the Pythagorean formula is introduced in most geometry texts. Well, it’s easy to calculate: a 3-4-5 triangle is a right triangle because 3-squared plus 4-squared, that is 9 + 16, is equal to 5-squared or 25.
The point of this article is to show some geometric examples of a 3-4-5 coming out of the blue. I’m not trying to give a comprehensive rundown. In working some problems offered by Presh Talwalkar at his website “Mind Your Decisions,” the 3-4-5 triangle kept showing up. I recommend this site for challenging problems, concise solutions, and illuminating videos. For space considerations, the proofs are concise – I offer my apologies.
In the first sketch we begin with a square and a semicircle. When we run a tangent line from an upper corner, the triangles show up. In the process of solving the length AF, what Presh was asking for, we discover that ADF is a 3-4-5. I made the side of the square measure 4 to get those numbers, but for any measures this configuration yields ADF in a 3-4-5 ratio.
Next, we begin with another square and quarter circles drawn from the bottom corners. The Pythagorean formula solves for the radius of three-eighths, but notice that the other measures in ABC are 4/8 and 5/8 putting the sides in a 3-4-5 ratio.
My last example comes from Universal Patterns [Boles and Newman] and it relates a 3-4-5 to an even more ubiquitous relation, the golden ratio. The authors did not provide a proof, but I was able to discover one. The 3-4-5 triangle shows up as half of a 6-5-5, drawn for the convenience of inscribing a circle. Line CA bisects angle C, resulting in the circle’s diameter AB and the extra bit BC to be in golden proportions.
Check out Presh Talwalkar’s site. His specialties are in game theory and probability in general.
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