Aztec dancer and ant

I took this photograph close to a waterfall off Harrogate Dr. in northwest Austin last year on 7/24. Whether the ant ran any risk of getting eaten by the Aztec dancer damselfly, Argia nahuana, I don’t know, but the date reminds me of something I do know, namely that 7 and 24 are the perpendicular sides of a 7-24-25 right triangle because 7 squared plus 24 squared equals 25 squared. Other right triangles with the shortest side an odd number are  5-12-13,  9-40-41,  11-60-61,  13-84-85,  and the familiar 3-4-5. Can you figure out how to get the two longer sides of each right triangle of this type if you know only the shortest side?

(Speaking of math, did anyone notice that the number 63 that played a role in yesterday’s post can be written in base 2 as 111111?)

© 2015 Steven Schwartzman