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10.1 Pythagorean Theorem

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Dee Black
Dee Black

The document provides instruction on the Pythagorean theorem. It defines key vocabulary like hypotenuse and leg. It explains how to use the theorem to find the length of an unknown side of a right triangle if two sides are known. Examples show setting up and solving equations using the theorem. The document also discusses using the theorem to determine if a triangle is right based on the lengths of its sides.

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10.1 Pythagorean Theorem I CAN: -SOLVE PROBLEMS USING THE PYTHAGOREAN THEOREM -IDENTIFY RIGHT TRIANGLES
Solve It! Pearson Warm Up
Flocab! http://www.flocabulary.com/pythagorean-theorem/
BIG Ideas The lengths of the sides of a right triangle have a special relationship.
BIG Ideas The lengths of the sides of a right triangle have a special relationship. If the lengths of any two sides of a right triangle are known, the length of the third side can be found.
BIG Ideas The lengths of the sides of a right triangle have a special relationship. If the lengths of any two sides of a right triangle are known, the length of the third side can be found. If the length of each side of a triangle is known, then whether the triangle is a right triangle can be determined.
FOCUS QUESTION: Why is the Pythagorean Theorem useful?
Vocabulary to Know Hypotenuse The side opposite the right angle
Vocabulary to Know Leg Each of the sides forming the right angle
Vocabulary to Know Pythagorean Theorem In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2
Finding the Hypotenuse and Leg Lengths You can use the Pythagorean Theorem to find the length of a right triangles hypotenuse given the lengths of its legs. Using the Pythagorean Theorem to solve for a side length involves finding a principal square root because side lengths are always positive.
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? 6 in. 6 in.
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 6 in. 6 in.
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 6 in. 62 + 62 = c2 6 in.
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 6 in. 62 + 62 = c2 Simplify 36 + 36 = c2 6 in.
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 6 in. 62 + 62 = c2 Simplify 36 + 36 = c2 6 in. Simplify 72 = c2
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 6 in. 36 + 36 = c2 Simplify 72 = c2 6 in. Find the principal square root. =c
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 36 + 36 = c2 6 in. Simplify 72 = c2 Find the principal square root. =c 6 in. Use a calculator 8.5 c Convert to a fraction!!
Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 36 + 36 = c2 Simplify 6 in. 72 = c2 Find the principal square root. =c Use a calculator 6 in. 8.5 c Convert to a fraction!! Write answer sentence. The hypotenuse is 8 遜 in.
Finding the hypotenuse Dry erase practice What is the length of the hypotenuse of a right triangle with leg lengths 9 cm and 12 cm?
Finding the Length of a Leg What is the side length b in the triangle? 5 cm 13 cm b
Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 5 cm 13 cm b
Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 5 cm 13 cm b
Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 b
Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate the variable by using the inverse operation. b
Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate the variable by using the inverse operation. Subtract 25 from each side. b2 = 144 b
Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate the variable by using the inverse operation. Subtract 25 from each side. b2 = 144 b Find the principal square root of each side. b = 72
Find the Length of a Leg Dry Erase Practice What is the side length a in the triangle? 12 a 15
Vocabulary to Know Conditional An if-then statement such as If an animal is a horse, then it has four legs.
Vocabulary to Know Hypothesis The if part of the conditional. Conclusion The then part of the conditional. Converse Switches the hypothesis and conclusion.
Conditional You can write the Pythagorean Theorem as a conditional: If a triangle is a right triangle with legs of lengths a and b and hypotenuse of length c, then a2 + b2 = c2. The converse of the Pythagorean Theorem is always true.
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 4 m, 8 m, 10 m 10 in., 24 in., 26 in. 8 ft., 15 ft., 16 ft.
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 62 + 242 ___ 252
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 62 + 242 ___ 252 36 + 576 ___ 625
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 62 + 242 ___ 252 36 + 576 ___ 625 612 625 NO
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 4 m, 8 m, 10 m 42 + 82 = 102
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 4 m, 8 m, 10 m 42 + 82 ___ 102 16 + 64 ___ 100
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 4 m, 8 m, 10 m 42 + 82 ___ 102 16 + 64 ___ 100 80 100 NO
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 10 in., 24 in., 26 in. 102 + 242 ___ 262 100 + 576 ___ 676
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 10 in., 24 in., 26 in. 102 + 242 ___ 262 100 + 576 ___ 676 676 = 676 YES
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 8 ft., 15 ft., 16 ft. 82 + 152 ___ 162
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 8 ft., 15 ft., 16 ft. 82 + 152 ___ 162 64 + 225 ___ 256
Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 8 ft., 15 ft., 16 ft. 82 + 152 ___ 162 64 + 225 ___ 256 289 256
Identifying Right Triangles Dry Erase Practice Could the lengths 20 mm, 47 mm, and 52 mm be the side lengths of a right triangle? Explain.
Focus Question Answer Why is the Pythagorean Theorem useful?
BIG Ideas The lengths of the sides of a right triangle have a special relationship. If the lengths of any two sides of a right triangle are known, the length of the third side can be found. If the length of each side of a triangle is known, then whether the triangle is a right triangle can be determined.
Assignment Pages 628-630 1-3 6-19 20-28 even 29-40

More Related Content

10.1 Pythagorean Theorem

  • 1. 10.1 Pythagorean Theorem I CAN: -SOLVE PROBLEMS USING THE PYTHAGOREAN THEOREM -IDENTIFY RIGHT TRIANGLES
  • 2. Solve It! Pearson Warm Up
  • 3. Flocab! http://www.flocabulary.com/pythagorean-theorem/
  • 4. BIG Ideas The lengths of the sides of a right triangle have a special relationship.
  • 5. BIG Ideas The lengths of the sides of a right triangle have a special relationship. If the lengths of any two sides of a right triangle are known, the length of the third side can be found.
  • 6. BIG Ideas The lengths of the sides of a right triangle have a special relationship. If the lengths of any two sides of a right triangle are known, the length of the third side can be found. If the length of each side of a triangle is known, then whether the triangle is a right triangle can be determined.
  • 7. FOCUS QUESTION: Why is the Pythagorean Theorem useful?
  • 8. Vocabulary to Know Hypotenuse The side opposite the right angle
  • 9. Vocabulary to Know Leg Each of the sides forming the right angle
  • 10. Vocabulary to Know Pythagorean Theorem In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2
  • 11. Finding the Hypotenuse and Leg Lengths You can use the Pythagorean Theorem to find the length of a right triangles hypotenuse given the lengths of its legs. Using the Pythagorean Theorem to solve for a side length involves finding a principal square root because side lengths are always positive.
  • 12. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? 6 in. 6 in.
  • 13. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 6 in. 6 in.
  • 14. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 6 in. 62 + 62 = c2 6 in.
  • 15. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 6 in. 62 + 62 = c2 Simplify 36 + 36 = c2 6 in.
  • 16. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 6 in. 62 + 62 = c2 Simplify 36 + 36 = c2 6 in. Simplify 72 = c2
  • 17. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 6 in. 36 + 36 = c2 Simplify 72 = c2 6 in. Find the principal square root. =c
  • 18. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 36 + 36 = c2 6 in. Simplify 72 = c2 Find the principal square root. =c 6 in. Use a calculator 8.5 c Convert to a fraction!!
  • 19. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 36 + 36 = c2 Simplify 6 in. 72 = c2 Find the principal square root. =c Use a calculator 6 in. 8.5 c Convert to a fraction!! Write answer sentence. The hypotenuse is 8 遜 in.
  • 20. Finding the hypotenuse Dry erase practice What is the length of the hypotenuse of a right triangle with leg lengths 9 cm and 12 cm?
  • 21. Finding the Length of a Leg What is the side length b in the triangle? 5 cm 13 cm b
  • 22. Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 5 cm 13 cm b
  • 23. Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 5 cm 13 cm b
  • 24. Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 b
  • 25. Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate the variable by using the inverse operation. b
  • 26. Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate the variable by using the inverse operation. Subtract 25 from each side. b2 = 144 b
  • 27. Finding the Length of a Leg What is the side length b in the triangle? Use the Pythagorean Theorem. a2 + b2 = c2 Substitute 5 for a and 13 for c. 52 + b2 = 132 Simplify. 5 cm 13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate the variable by using the inverse operation. Subtract 25 from each side. b2 = 144 b Find the principal square root of each side. b = 72
  • 28. Find the Length of a Leg Dry Erase Practice What is the side length a in the triangle? 12 a 15
  • 29. Vocabulary to Know Conditional An if-then statement such as If an animal is a horse, then it has four legs.
  • 30. Vocabulary to Know Hypothesis The if part of the conditional. Conclusion The then part of the conditional. Converse Switches the hypothesis and conclusion.
  • 31. Conditional You can write the Pythagorean Theorem as a conditional: If a triangle is a right triangle with legs of lengths a and b and hypotenuse of length c, then a2 + b2 = c2. The converse of the Pythagorean Theorem is always true.
  • 32. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 4 m, 8 m, 10 m 10 in., 24 in., 26 in. 8 ft., 15 ft., 16 ft.
  • 33. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 62 + 242 ___ 252
  • 34. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 62 + 242 ___ 252 36 + 576 ___ 625
  • 35. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 6 in., 24 in., 25 in. 62 + 242 ___ 252 36 + 576 ___ 625 612 625 NO
  • 36. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 4 m, 8 m, 10 m 42 + 82 = 102
  • 37. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 4 m, 8 m, 10 m 42 + 82 ___ 102 16 + 64 ___ 100
  • 38. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 4 m, 8 m, 10 m 42 + 82 ___ 102 16 + 64 ___ 100 80 100 NO
  • 39. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 10 in., 24 in., 26 in. 102 + 242 ___ 262 100 + 576 ___ 676
  • 40. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 10 in., 24 in., 26 in. 102 + 242 ___ 262 100 + 576 ___ 676 676 = 676 YES
  • 41. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 8 ft., 15 ft., 16 ft. 82 + 152 ___ 162
  • 42. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 8 ft., 15 ft., 16 ft. 82 + 152 ___ 162 64 + 225 ___ 256
  • 43. Identifying Right Triangles Which set of lengths could be the side lengths of a right triangle? 8 ft., 15 ft., 16 ft. 82 + 152 ___ 162 64 + 225 ___ 256 289 256
  • 44. Identifying Right Triangles Dry Erase Practice Could the lengths 20 mm, 47 mm, and 52 mm be the side lengths of a right triangle? Explain.
  • 45. Focus Question Answer Why is the Pythagorean Theorem useful?
  • 46. BIG Ideas The lengths of the sides of a right triangle have a special relationship. If the lengths of any two sides of a right triangle are known, the length of the third side can be found. If the length of each side of a triangle is known, then whether the triangle is a right triangle can be determined.
  • 47. Assignment Pages 628-630 1-3 6-19 20-28 even 29-40

Editor's Notes

  • #3: You must have a Pearson Log In to View this.
  • #4: You must have a flocabulary account to log in here.
  • #8: Dont have students answer the question at this point. This is just here to get their thinking started.
  • #21: 15 cm
  • #29: 9
  • #45: No. 20 squared + 47 squared = 2609. 52 squared = 2704.
  • #48: This assignment is for Algebra I Foundation by Pearson. It was the recommended assignment for a block schedule. I usually end up altering assignments since I teach Special Education.