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Teacher Key Geometry Right Triangles Use the Pythagorean theorem to find the length of the hypotenuse. Show your work. If the answer is not a perfect square, leave your answer in square root form. 13 cm x = _____ x 5 cm A. 15 inches B. 20 feet c y C. 8 inches 12 cm 17 inches c = _____ 82 + 152 = c2 64 + 225 = c2 289 = c2 17 = c 29 feet y = _____ 202 + 212 = y2 400 + 441 = y2 841 = y2 29 = y 53 cm a = _____ 22 + 72 = a2 4 + 49 = a2 53 = a2 53 = a 21 feet a 2 cm D. 7 cm © 2003 CompassLearning, Inc. 52 + 122 = x2 25 + 144 = x2 169 = x2 13 = x Activity 67252 Teacher Key Geometry Right Triangles 16 feet 20 feet h = _____ 12 feet E. h 5 inches F. k 61 inches k = _____ 162 + 122 = h2 256 + 144 = h2 400 = h2 20 = h 62 + 52 = k2 36 + 25 = k2 61 = k2 61 = k 6 inches c G. 6 cm 10 c = _____ cm 8 cm 2 inches H. 3 inches w © 2003 CompassLearning, Inc. 82 + 62 = c2 64 + 36 = c2 100 = c2 10 = c 11 w = _____ inches ( 2 )2 + 32 = w2 2 + 9 = w2 11 = w2 11 = w Activity 67252 Teacher Key Geometry Right Triangles Extension The Pythagorean theorem can also be used to find the leg of a right triangle when the length of the hypotenuse and the other leg are known. 26 cm b 10 cm Subtract the value of the squared leg from both sides to isolate the unknown value. a2 + b2 = c2 a – a2 + b2 = c2 – a2 0 + b2 = c2 – a2 b2 = c2 – a2 2 Substitute the known values into the formula. a = 10 cm c = 26 cm b2 = c2 – a2 b2 = 262 – 102 Find the squares of the numbers. b2 = 262 – 102 b2 = 676 – 100 Find the difference. b2 = 676 – 100 b2 = 576 Find the square root of the difference to solve for the variable. b2 = 576 2 b = 576 b = 24 © 2003 CompassLearning, Inc. Activity 67252 Teacher Key Geometry Right Triangles Use the Pythagorean theorem to find the length of the missing leg. Show your work. If the answer is not a perfect square, leave your answer in square root form. I. 9 cm b = _____ 15 cm b 12 cm J. 12 inches K. 17 feet 6 inches b a 108 inches b = _____ 62 + b2 = 122 b2 = 122 - 62 b2 = 144 - 36 b2 = 108 b = 108 8 feet a = _____ a2 + 152 = 172 a2 = 172 – 152 a2 = 289 – 225 a2 = 64 a=8 36 cm a = _____ a2 + 152 = 392 a2 = 392 – 152 a2 = 1521 – 225 a2 = 1296 a = 36 15 cm 15 feet L. © 2003 CompassLearning, Inc. 39 cm a 122 + b2 = 152 b2 = 152 – 122 b2 = 225 – 144 b2 = 81 b =9 Activity 67252 Teacher Key Geometry Connections Think About It Using what you know about the Pythagorean theorem, explain how you could tell if, given the lengths of three sides of a triangle, that triangle was a right triangle. SAMPLE RESPONSE: The Pythagorean theorem says that the sum of the squares of ________________________________________________________________________ the two legs is equal to the square of the hypotenuse. So, if you have three numbers ________________________________________________________________________ that represent the three side lengths of a triangle, you could square all three of them ________________________________________________________________________ and then check to see if two of the squared numbers add up to the third. If so, the ________________________________________________________________________ assumption can be made that the original three side lengths represented the lengths ________________________________________________________________________ of the sides of a right triangle. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Pythagorean Triple A Pythagorean triple consists of three whole numbers that satisfy the equation a2 + b2 = c2, where c is the largest number. An example is the triple 3-4-5, where 32 + 42 = 52. Experiment with whole numbers to generate a list of as many Pythagorean triples as you can discover. Sample Response: There are an infinite number of Pythagorean triples. 3-4-5 6-8-10 5-12-13 8-15-17 12-16-20 17-24-25 10-24-26 © 2003 CompassLearning, Inc. Activity 67252