Notes for the week of October 5 through 8

Classification of triangles, including the triangle inequality and Heron's formula, which will give us the area of a triangle defined by side lengths, here using the letters u, v and w.

Finding a line segment of the length of the square root of any number.

The easiest thing is to find the square root of any given odd number. To find the square root of any given even number, find the square root of the odd number one less than it. For example, if I want to find sqrt(44), it will be a two step process, where first we find the sqrt(43).



Step 1: 43/2 = 21.5, so the whole numbers closest to splitting 43 in half are 22 and 21. It's easy to prove that 22² - 21² = 43, so we make a triangle with the hypotenuse = 22 and the long leg = 21.  The short leg will be sqrt(43).

Step 2: Now we make a new right triangle by extending the leg of length 21 by 1. Now we have a new right triangle with legs of length sqrt(43) and 1, and the hypotenuse will be sqrt(44).