Right Triangle Day, September 12, 2015

Why Did I Name September 12, 2015 "Right Triangle Day"?

In Right Triangles (triangles with a 90 degree angle), the sum of the squares of the legs (the sides of the triangle that make up the right angle) is equal to the square of the hypotenuse (the side opposite the right angle or the longest side of a right triangle). There are a few "standard" right triangles, the most "famous" is the 3-4-5 triangle (3 squared + 4 squared = 5 squared or 9 + 16 = 25). This works on triangles where you multiply all the numbers by 2, or, for example, 6-8-10 (6 squared + 8 squared = 10 squared or 36 + 64 = 100).

September 12, 2015 or 

9-12-15

Another Standard Right Triangle is the 9-12-15 triangle (3 X 3,4,5 triangle) because 81 + 144 = 225 (9 squared + 12 squared = 15 squared).

Thank Pythagoras

Pythagoras was an Ancient Greek Mathematician. His theorem helps us to understand right triangles (a right triangle is a triangle with a 90 degree angle in it -- a 90 degree angle is an angle where the two sides of the angle are perpendicular to each other). He proved that the sum of the squares of the two legs (the two sides that make up the right angle) of the right triangle is the square of the hypotenuse (the side of the triangle opposite the right angle).

I was thinking about Pi day a couple of days beforehand (on March 12, 2013) and I realized that in two months (May 12, 2013) would be the numbers of one of the standard whole rational number right triangles (oftentimes, the sum of the squares of the two legs of a triangle do not add up to a number that is a perfect square (1, 4, 9, 16, 25, to name a few -- go to my How to Figure Square Roots blog to see a list of the perfect squares from 1 to 625 -- 25 squared). So we math geeks try to remember some of the combinations that ARE perfect squares:

3, 4, 5
5,12,13
8,15,17

to name a few (keep in mind, multiplying each side by the same number -- like 2 X 3, 4, 5 is a 6, 8, 10 triangle and 3 X 5, 12, 13 is a 15, 36, 39 triangle -- 6, 8, 10 would look like 36 + 64 = 100 and the 15, 36, 39 triangle would look like 225 + 1296 = 1521).

So Happy Right Triangle Day to all my Friends. This day won't come again until 2115 (though August 15, 2017, December 16, 2020 and October 24, 2026 might work also, as did March 4, 2005 and May 12, 2013!).

Math Proofs

When I was younger, I sort of assumed that if there were combinations where a squared + b squared = c squared that there must be plenty of combinations where a cubed + b cubed = c cubed or a to the fourth + b to the fourth = c to the fourth, etc. BUT.....

I found out years later that mathematicians suspect that a squared + b squared = c squared is the ONLY exponent number that this works with rational numbers in all the spots (rational numbers are numbers that can be written as whole numbers or whole numbers with fractions or decimals that end -- examples of irrational numbers are Pi, the square root of 2 and the cube root of 7 -- these are numbers that can only be approximated).