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Name _______________________________________ Date ___________________ Class __________________ Section 5.7 The Pythagorean Theorem The Pythagorean Theorem states that the following relationship exists among the lengths of the legs, a and b, and the length of the hypotenuse, c, of any right triangle. a2 + b2 = c2 Use the Pythagorean Theorem to find the value of x in each triangle. a2 + b2 = c2 Pythagorean Theorem a2 + b2 = c2 x2 + 62 = 92 Substitute. x2 + 42 = (x + 2)2 x2 + 36 = 81 Take the squares. x2 + 16 = x2 + 4x + 4 x2 = 45 x= 45 x= 3 5 4x = 12 Simplify. x=3 Take the positive square root and simplify. Find the value of x. Give your answer in simplest radical form. 1. 2. ________________________________________ 3. ________________________________________ 4. ________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry Name _______________________________________ Date __________________ Class __________________ Section 5.7 The Pythagorean Theorem continued A Pythagorean triple is a set of three nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2. Pythagorean Triples Not Pythagorean Triples 3, 4, 5, 5, 12, 13 2, 3, 4 6, 9, 117 You can use the following theorem to classify triangles by their angles if you know their side lengths. Always use the length of the longest side for c. Pythagorean Inequalities Theorem m∠C < 90° m∠C > 90° If c2 > a2 + b2, then ∆ABC is obtuse. If c2 < a2 + b2, then ∆ABC is acute. Consider the measures 2, 5, and 6. They can be the side lengths of a triangle since 2 + 5 > 6, 2 + 6 > 5, and 5 + 6 > 2. If you substitute the values into c2 < a2 + b2, you get 36 > 29. Since c2 > a2 + b2, a triangle with side lengths 2, 5, and 6 must be obtuse. Find the missing side length. Tell whether the side lengths form a Pythagorean triple. Explain. 5. 6. Tell whether the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7. 4, 7, 9 ________________________ 10. 9, 12, 15 ________________________ 8. 10, 13, 16 ________________________ 11. 5, 14, 20 ________________________ 9. 8, 8, 11 ________________________ 12. 4.5, 6, 10.2 ________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry Name _______________________________________ Date __________________ Class __________________ Section 5.7 1. x = 12 2. x = 3. x = 4. x = 40 39 29 5. 10; yes; the three side lengths are nonzero whole numbers that satisfy a2 + b2 = c2. 6. 95; no; 7. yes; obtuse 9. yes; acute 11. no 95 is not a whole number. 8. yes; acute 10. yes; right 12. yes; obtuse Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry