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).

How to Figure "Square Roots" of Numbers that are not Perfect Squares

This is not a post about using a calculator to find a square root. This is not a post about finding a square root to its nearest 10th or 100th or 2 places. This is a lens that helps you get from a large number's square root to a smaller number's square root.

In other words, this will show you how to go from ([square root] 20) to (2 ([square root] 5).

Reducing Irrational Square Roots

One of the sorts of problems that shows up on Algebra tests is taking a square root of a number that isn't a perfect square (see a list of all the integers that are perfect squares from 1-625 at the bottom of this lens) and reducing it to an easier to handle version. For example, in the picture here, we start with [square root] 98. But, if you notice when you look at the list, 98 is NOT a perfect square.

98, however, equals 2 X 49. 49 <b>is</b> a perfect square -- it's 7 squared (or 7 X 7). So, as you can see in the second step, [square root] 98 = [square root] 49 X [square root] 2. Or, more succinctly, 7 X [square root] 2.

If the number you are trying to divide is one you don't know (as in you can't tell right away by looking at it what perfect square might be a factor), check out <a href="http://www.squidoo.com/LowCommonDenom">Lowest Common Denominator</a> to see how to break numbers into their factors. Or, if the number is low enough, you can look at the list of squares from 1-625 further down in this lens and work from there.

Perfect Squares

This is a list of the first 25 positive integers with their squares.

1. 1
2. 4
3. 9
4. 16
5. 25
6. 36
7. 49
8. 64
9. 81
10. 100
11. 121
12. 144
13. 169
14. 196
15. 225
16. 256
17. 289
18. 324
19. 361
20. 400
21. 441
22. 484
23. 529
24. 576
25. 625