8.1 The Pythagorean Theorem and Its Converse We will learn to use the Pythagorean Theorem and its converse.

The Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. B AC c a b If ∆ABC is a right triangle Then… leg² + leg² = hypotenuse² a² + b² = c²

Vocabulary Pythagorean Triple: is a set of nonzero whole numbers a, b, and c that make the equation a² + b² = c² true. – Common Pythagorean Triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 – If you multiply each number in a Pythagorean Triple by the same whole number, the three numbers that result also form a Pythagorean Triple. 3, 4, 5  6, 8, 103, 4, 5  9, 12, 15

Finding the Lengths of the Hypotenuse Example: What is the length of the hypotenuse of ∆ABC? Do the side lengths of ∆ABC form a Pythagorean Triple? Explain? A B C leg² + leg² = hypotenuse² 20² + 21² = c² = c² 841 = c² √(841) = c 29 = c The side lengths 20, 21, and 29 form a Pythagorean Triple because they are whole numbers that satisfy a² + b² = c²

Finding the Length of the Hypotenuse You Try: The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? Do these side lengths form a Pythagorean Triple? 10² + 24² = c² = c² 676 = c² √(676) = c 26 = c Yes, side lengths 10, 24, and 26 are whole numbers that satisfy a² + b² = c²

Finding the Length of a Leg Example: What is the value of x? Express your answer in simplest radical form. 20 x 8 a² + b² = c² x² + 8² = 20² x² + 64 = 400 x² = 336 x = √(336) x = √(16·21) x = 4√(21)

Finding the Length of a Leg You Try: The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form. a² + b² = c² x² + 6² = 12² x² + 36 = 144 x² = 108 x = √(108) x = √(36·3) x = 6√(3)

Finding Distance Example: Dog agility courses often contain a seesaw obstacle, as shown below. To the nearest inch, how far above ground are the dog’s paws when the seesaw is parallel to the ground? 36 in. 26 in. a² + b² = c² x² + 26² = 36² x² = 1296 x² = 620 x = √(620) x = √(4·155) x = 2√(155) or ≈ The dog’s paws are 25 inches from the ground when the seesaw is parallel to the ground.

Finding Distance You Try: The size of a computer monitor is the length of its diagonal. You want to buy a 19- inch monitor that has a height of 11 inches. What is the width of the monitor? Round to the nearest tenth of an inch a² + b² = c² x² + 11² = 19² x² = 361 x² = 240 x = √(240) x = √(16·15) x = 4√(15) or ≈ 15.5 The monitor is 15.5 inches wide.

The Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. B AC c a b If a² + b² = c² Then… ∆ABC is a right triangle.

Identifying a Right Triangle Example: A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain. a² + b² = c² 13² + 84² = 85² = = 7225 Yes, the triangle is a right triangle because 13² + 84² = 85². ** The longest side of the triangle always needs to be plugged in for c, the hypotenuse.

Identifying a Right Triangle You Try: A right triangle has side lengths 16, 48, and 50. Is the triangle a right triangle? Explain. a² + b² = c² 16² + 48² = 50² = ≠ 2500 No, the triangle is not a right triangle because 16² + 48² ≠ 50².

Pythagorean Inequalities Theorem If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. A B C c b a If… c² > a² + b² Then… ∆ABC is obtuse

Pythagorean Inequalities Theorem If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. A B C c b a If… c² < a² + b² Then… ∆ABC is acute.

Classifying a Triangle Example: A triangle has side lengths 6, 11, and 14. Is it acute, obtuse, or right? c² > a² + b² 14² > 6² + 11² 196 > > 157 This triangle is obtuse since 14² > 6² + 11².

Classifying a Triangle You Try: Is a triangle with side lengths 7, 8, and 9 acute, obtuse, or right? c² < a² + b² 9² < 7² + 8² 81 < < 113 This triangle is acute since 9² < 7² + 8².

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