5.7A The Pythagorean Theorem Objectives: G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. For the Board: You will be able to use the Pythagorean Theorem and its converse and its inequalities to solve problems. Anticipatory Set: A Pythagorean Triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2. 3, 4, 5 is an example of a Pythagorean Triple. 32 + 42 = 9 + 16 = 25 = 52 0.5, 1.2, 1.3 is not an example of a Pythagorean Triple, even through 0.52 + 1.22 = 1.32, because these numbers are not positive integers. 2, 3, 4 is not an example of a Pythagorean Triple, because 22 + 32 ≠ 42. White Board Activity: Practice: Which of the following are Pythagorean Triples? a. 5, 12, 13 b. 4, 5, 9 c. 8, 15, 17 52 + 122 = 25 + 144 = 169 = 132 Yes 42 + 52 = 16 + 25 = 41 which is not 92 (81) No 82 + 152 = 64 + 225 = 289 = 172 Yes Instruction: Open the book to page 362 and read example 3. Example: Find the missing side length, then tell if the side lengths form a Pythagorean triple. a. b. 14 4 12 48 142 + 482 = 196 + 2304 = c2 42 + b2 = 122 16 + b2 = 144 c2 = 2500 c = 50 b2 = 128 b = 128 264 8 2 Yes No White Board Activity: Practice: Find the missing side length, then tell if the side lengths form a Pythagorean triple. a. b. x2 + 242 = 262 10 8 24 x2 = 576 = 676 x2 = 100 26 82 + 102 = x2 x = 10 64 + 100 = x2 yes 164 = x2 x = 12.8 No Pythagorean Inequalities Theorem In ΔABC, c is the length of the longest side. If c2 < a2 + b2, then ΔABC is an acute triangle. If c2 = a2 + b2, then ΔABC is a right triangle. If c2 > a2 + b2, then ΔABC is an obtuse triangle. c c a c a a b b b Recall: Triangle Inequality Theorem The sum of the measures of the two smaller sides must be larger than the measure of the third side. Example: 4, 8, 9 form a triangle because 4 + 8 = 12 > 9 3, 4, 7 do not form a triangle because 3 + 4 = 7 which is not > 7 2, 5, 12 do not form a triangle because 2 + 5 = 7 which in not > 12 Open the book to page 363 and read example 4. Example: Tell if the measures con be the side lengths of a triangle. If so classify the triangle as acute, obtuse, or right. A. 7, 8, 9 B. 8, 8, 15 7 + 8 = 15 > 9 8 + 8 = 16 > 15 Yes, triangle. Yes, triangle. 92 72 + 82 152 82 + 82 81 49 + 64 225 64 + 64 81 113 225 128 81 < 113, the triangle is acute. 225 > 128,the triangle is obtuse. White Board Activity: Practice: 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 5 + 7 = 12 > 10 5 + 8 = 13 < 17 Yes they form a triangle No does not form a triangle 102 52 + 72 100 25 + 49 = 74 Since 100 > 74 the triangle is obtuse Since practice prob. b does not form a triangle, change one of the sides so that it does form a triangle. Then determine whether it is acute, obtuse, or right. Example: Change the 8 to 15, then 5 + 15 = 20 which is > 17. 172 52 + 152 289 25 + 225 = 250 Since 289 > 250 the triangle is obtuse. Assessment: Question student pairs. Independent Practice: Text: pgs. 364 - 367 prob. 6 – 14, 19 – 27. For a Grade: Text: pgs. 364 – 367 prob. 8, 10, 20, 22. .