Section 6: The Pythagorean Theorem

# Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 9.6 The Pythagorean Theorem 959 Version: Fall 2007 9.6 The Pythagorean Theorem Pythagoras. Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in south- ern Italy called Crotone, whose members eventually became known as the Pythagoreans. The inner circle at the school, the Mathematikoi , lived at the school, rid themselves of all personal possessions, were vegetarians, and observed a strict vow of silence. They studied mathematics, philosophy, and music, and held the belief that numbers constitute the true nature of things, giving numbers a mystical or even spiritual quality. Today, nothing is known of Pythagoras’s writings, per- haps due to the secrecy and silence of the Pythagorean so- ciety. However, one of the most famous theorems in all of mathematics does bear his name, the Pythagorean Theorem. Pythagorean Theorem. Let c represent the length of the hypotenuse , the side of a right triangle directly opposite the right angle (a right angle measures 90 ) of the triangle. The remaining sides of the right triangle are called the legs of the right triangle, whose lengths are designated by the letters a and b . a b c The relationship involving the legs and hypotenuse of the right triangle, given by a 2 + b 2 = c 2 , (1) is called the Pythagorean Theorem . Note that the Pythagorean Theorem can only be applied to right triangles. Let’s look at a simple application of the Pythagorean Theorem ( 1 ). l⚏ Example 2. Given that the length of one leg of a right triangle is 4 centimeters and the hypotenuse has length 8 centimeters, find the length of the second leg. Let’s begin by sketching and labeling a right triangle with the given information. We will let x represent the length of the missing leg. Copyrighted material. See: 1

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960 Chapter 9 Radical Functions Version: Fall 2007 x 4 cm 8 cm Figure 1. A sketch makes things a bit easier. Here is an important piece of advice. Tip 3. The hypotenuse is the longest side of the right triangle. It is located di- rectly opposite the right angle of the triangle. Most importantly, it is the quantity that is isolated by itself in the Pythagorean Theorem. a 2 + b 2 = c 2 Always isolate the quantity representing the hypotenuse on one side of the equa- tion. The legs go on the other side of the equation. So, taking the tip to heart, and noting the lengths of the legs and hypotenuse in Figure 1 , we write 4 2 + x 2 = 8 2 . Square, then isolate x on one side of the equation. 16 + x 2 = 64 x 2 = 48 Normally, we would take plus or minus the square root in solving this equation, but x represents the length of a leg, which must be a positive number. Hence, we take just the positive square root of 48.
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