C CHAPTER The Pythagorean Theorem

Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles.

Pythagorean Theorem The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in previous lessons, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a 2 + b 2 = c 2

Proving the Pythagorean Theorem Pythagoras of Samos, c.560–480 BC, was a Greek philosopher and religious leader who was responsible for important developments in the history of mathematics, astronomy, and the theory of music. He migrated to Croton where he founded a philosophical and religious school that attracted many followers. Because no reliable contemporary records survive, and because the school practiced both secrecy and communalism, the contributions of Pythagoras himself and those of his followers cannot be distinguished. The most important discovery of this school was the fact that the diagonal of a square is not a rational multiple of its side (that is, the diagonal of a square is not a number that can be expressed as the ratio of two whole numbers.). I n essence, this showed the existence of irrational numbers. This discovery disturbed Greek mathematicians and the Pythagoreans themselves, who believed that whole numbers and their ratios could account for geometrical properties. Pythagoreans believed that all relations could be reduced to number relations ("all things are numbers"). The Pythagoreans knew, as did the Egyptians before them, that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. The so-called Pythagorean theorem, that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, may have been known in Babylonia, where Pythagoras traveled in his youth. The Pythagoreans, however, are usually credited with the first proof of this theorem. Much of the Pythagorean doctrine that has survived consists of numerology and number mysticism, and the influence of the belief that the world can be understood through mathematics. That belief was extremely important to the development of science and mathematics.

Proving the Pythagorean Pro

Example Find the value of x. Give your answer in simplest radical form.

Example Find the value of x. Give your answer in simplest radical form.

Example Find the value of x. Give your answer in simplest radical form.

Application Randy is building a rectangular picture frame. He wants the ratio of the length to the width to be 3:1 and the diagonal to be 12 centimeters. How wide should the frame be? Round to the nearest tenth of a centimeter.

Application What if...? According to the recommended safety ratio of 4:1, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch.

A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple.

Example Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

Example Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

Converse of the Pythagorean theorem The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

Pythagorean You can also use side lengths to classify a triangle as acute or obtuse. A B C c b a

To understand why the Pythagorean inequalities are true, consider ∆ABC.

Example Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. A. 5, 7, 10 B. 5, 8, 17

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Student Guided Practice Do problems 2-9 in your book page 364

Homework Do problems in your book page 365

closure Today we learned about Pythagorean theorem Next class we are going to apply special right triangles