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5-3A The Pythagorean Theorem

5-3A The Pythagorean Theorem. You used the Pythagorean Theorem to develop the Distance Formula. Use the Pythagorean Theorem. Use the Converse of the Pythagorean Theorem. c. a. b. Pythagorean Theorem.

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5-3A The Pythagorean Theorem

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  1. 5-3A The Pythagorean Theorem You used the Pythagorean Theorem to develop the Distance Formula. • Use the Pythagorean Theorem. • Use the Converse of the Pythagorean Theorem.

  2. c a b Pythagorean Theorem The Pythagorean Theorem is used to calculate the length of any side of a right triangle when the lengths of the other two sides are known. Which side is the hypotenuse? Which sides are the legs?

  3. c a Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a2 + b2 = c2 b p. 547

  4. p. 547

  5. S T U Find the length of the third side of the right triangle ∆STU. ST=3, TU = 4 ST2 + TU2 = SU2 32 + 42 = SU2 9 + 16 = SU2 25 = SU2 5 = SU

  6. Answer: A. Find x. The side opposite the right angle is the hypotenuse, so c = x. a2 + b2 = c2 Pythagorean Theorem 42 + 72 = c2a = 4 and b = 7 65 = c2 Simplify. Take the positive square root of each side.

  7. Answer: B. Find x. The hypotenuse is 12, so c = 12. a2 + b2 = c2 Pythagorean Theorem x2 + 82 = 122b = 8 and c = 12 x2 + 64 = 144 Simplify. x2 = 80 Subtract 64 from each side. Take the positive square root of each side and simplify.

  8. S T U Find the length of the third side of the right triangle ∆STU. ST=7, SU = 10 ST2 + TU2 = SU2 72 + TU2 = 102 49 + TU2 = 102 49 + TU2 = 100 TU2 = 51

  9. A. B. C. D. A. Find x.

  10. A. B. C. D. B. Find x.

  11. Pythagorean Triple Pythagorean Triples are special sets of numbers that all the numbers are positive integers. A Pythagorean triple is 3, 4, and 5. 32 + 42 = 52 9 + 16 = 25

  12. Pythagorean Triples under 100 (3, 4, 5)( 5, 12, 13)( 7, 24, 25)( 8, 15, 17) ( 9, 40, 41)(11, 60, 61)(12, 35, 37) (13, 84, 85)(16, 63, 65)(20, 21, 29) (28, 45, 53)(33, 56, 65)(36, 77, 85) (39, 80, 89)(48, 55, 73)(65, 72, 97) Formula: Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 2mn, and n2 + m2 is a Pythagorean triple.

  13. p. 548

  14. 12 25 7 8 13 6 Find the length of the missing side. 5 Triple = 5, 12, 13 24 Triple = 7, 24, 25 10 Triple = 6, 8, 10

  15. ? Check: 242 + 102 = 262Pythagorean Theorem Use a Pythagorean triple to find x. Explain your reasoning. Notice that 24 and 26 are multiples of 2: 24 = 2 ● 12 and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 ● 5 or 10. Answer: x = 10  676 = 676Simplify.

  16. Use a Pythagorean triple to find x. A. 10 B. 15 C. 18 D. 24

  17. If you have the lengths of three sides of a triangle, you can use the converse of the Pythagorean Theorem to prove it is a right triangle. p. 550

  18. You can also use side lengths to classify an acute or obtuse triangle. p. 550

  19. ? c2 = a2 + b2 Compare c2 and a2 + b2. ? 152 = 122 + 92 Substitution A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 9 + 12 > 15  9 + 15 > 12  12 + 15 > 9  The side lengths 9, 12, and 15 can form a triangle. Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. 225 = 225 Simplify and compare. Answer:Since c2 = a2 + b2, the triangle is a right triangle.

  20. ? c2 = a2 + b2 Compare c2 and a2 + b2. ? 132 = 112 + 102 Substitution B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 10 + 11 > 13  10 + 13 > 11  11 + 13 > 10  The side lengths 10, 11, and 13 can form a triangle. Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. 169 < 221 Simplify and compare. Answer:Since c2 < a2 + b2, the triangle is acute.

  21. A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. A. yes, acute B. yes, obtuse C. yes, right D. not a triangle

  22. What is the Pythagorean Theorem? a2 + b2 = c2 • Why is it important? It is used to calculate the length of any side of a right triangle when the lengths of the other two sides are known. • What is a Pythagorean Triple? Pythagorean Triples are special sets of numbers that all the numbers are positive integers.

  23. 8-2 Assignment Page 552, 8-28 even

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