27 1 The The Pythagorean Theorem Warm Up

27. 1 The The. Pythagorean. Theorem Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Geometry Holt

27. 1 The Pythagorean Theorem Warm Up Classify each triangle by its angle measures. 1. 2. acute 3. Simplify right 12 4. If a = 6, b = 7, and c = 12, find a 2 + b 2 and find c 2. Which value is greater? 85; 144; c 2 Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Vocabulary Pythagorean triple Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1 -6, 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 Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 1 Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem 22 + 62 = x 2 Substitute 2 for a, 6 for b, and x for c. 40 = x 2 Simplify. Find the positive square root. Simplify the radical. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 2 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. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4 x is the distance in feet from the top of the ladder to the base of the wall. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 2 Continued a 2 + b 2 = c 2 (4 x)2 + x 2 = 302 17 x 2 = 900 Pythagorean Theorem Substitute 4 x for a, x for b, and 30 for c. Multiply and combine like terms. Since 4 x is the distance in feet from the top of the ladder to the base of the wall, 4(7. 28) 29 ft 1 in. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 3 Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a 2 + b 2 = c 2 42 + b 2 = 122 b 2 = 128 Pythagorean Theorem Substitute 4 for a and 12 for c. Multiply and subtract 16 from both sides. Find the positive square root. The side lengths do not form a Pythagorean triple because is not a whole number. Holt Mc. Dougal Geometry

27. 1 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. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem You can also use side lengths to classify a triangle as acute or obtuse. B c A Holt Mc. Dougal Geometry a b C

27. 1 The Pythagorean Theorem Remember! By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 4 Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 7, 10 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 5, 7, and 10 can be the side lengths of a triangle. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 4 Step 2 Classify the triangle. c 2 102 ? = a 2 + b 2 ? = 52 + 72 ? Compare c 2 to a 2 + b 2. Substitute the longest side for c. 100 = 25 + 49 Multiply. 100 > 74 Add and compare. Since c 2 > a 2 + b 2, the triangle is obtuse. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 5 Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 8, 17 Step 1 Determine if the measures form a triangle. Since 5 + 8 = 13 and 13 > 17, these cannot be the side lengths of a triangle. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 6 Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 3. 8, 4. 1, 5. 2 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 3. 8, 4. 1, and 5. 2 can be the side lengths of a triangle. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Example 6 Step 2 Classify the triangle. c 2 5. 22 ? = a 2 + b 2 ? = 3. 82 + 4. 12 Compare c 2 to a 2 + b 2. Substitute the longest side for c. ? 27. 04 = 14. 44 + 16. 81 Multiply. 27. 04 < 31. 25 Add and compare. Since c 2 < a 2 + b 2, the triangle is acute. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Lesson Quiz: Part I 1. Find the value of x. 12 2. An entertainment center is 52 in. wide and 40 in. high. Will a TV with a 60 in. diagonal fit in it? Explain. Holt Mc. Dougal Geometry

27. 1 The Pythagorean Theorem Lesson Quiz: Part II 3. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 13; yes; the side lengths are nonzero whole numbers that satisfy Pythagorean’s Theorem. 4. Tell if the measures 7, 11, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. yes; obtuse Holt Mc. Dougal Geometry