Click here to purchase the entire book in PDF format. Right Triangles I'll assume at this point that you know what a triangle is. If you do not understand what a triangle is, then I would recommend backing up a bit from this textbook and reading other tomes such as Trevor Draws a Triangle and the immortal classic, Baby's First Book of Euclidian Geometry. (Okay, okay... I lied... Those books don't really exist... I hope that this hasn't put a dent in our relationship, and that you can still trust me for the rest of this book...) Once upon a time, a Greek by the name of Pythagoras had a minor obsession with triangles. 2.1 Interestingly, Pythagoras, like many other Greeks of his time, recognized the direct link between mathematics and musical acoustics, so you'll see his name popping up all over the place as we go through this book. Anyways, back to triangles. The first thing that we have to define is something called a right triangle. This is just a regular old everyday triangle with one specific characteristic. One of its angles is a right angle meaning that it's $90^\circ$ as is shown in Figure 1.1. One other new word to learn. The side opposite the right angle (in Figure 1.1, that would be side $a$) is called the hypotenuse of the triangle. Figure 1.1: A right trangle with sides of lengths $a$, $b$ and $c$. Note that side $a$ is called the hypotenuse of the triangle. One of the things Pythagoras discovered was that if you take a right trangle and make a square from each of its sides as is shown in Figure 1.2, then the sum of the areas of the two smaller squares is equal to the area of the big square. Figure 1.2: Three squares of areas $A$, $B$ and $C$ created by making squares out of the sides of a right trangle of arbitrary dimensions. $A = B+C$ So, looking at Figure 1.2, then we can say that $A = B+C$. We also should know that the area of a square is equal to the square of the length of one of its sides. Looking at Figures 1.1 and 1.2 this means that $A = a^2$, $B = b^2$, and $C = c^2$. Therefore, we can put this information together to arrive at a standard equation for right triangles known as the Pythagorean Theorem, shown in Equation 1.3. $$a^2 = b^2 + c^2$$ (2.2) and therefore $$a = \sqrt{b^2 + c^2}$$ (2.3) This equation is probably the single most important key to understanding the concepts presented in this book, so you'd better remember it.