10.1 Pythagorean Theorem

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  • 1. 10.1 PythagoreanTheoremI 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 thePythagorean 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 LegLengths 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 = c26 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 thehypotenuse 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 = c26 in. Find the principal square root. =c 18. Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of thehypotenuse of the right triangle shown? Use the Pythagorean Theorem a2 + b2 = c2 Substitute 6 for a and b. 62 + 62 = c2 Simplify 36 + 36 = c26 in. Simplify 72 = c2 Find the principal square root. =c6 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 theright 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 calculator6 in. 8.5 c Convert to a fraction!! Write answer sentence. The hypotenuse is 8 in. 20. Finding the hypotenuse Dry erasepractice 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 cmb 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 cmb 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 = 169b 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 cm13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate thevariable 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 cm13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate thevariable by using the inverse operation. Subtract 25 from each side. b2 = 144b 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 cm13 cm 25 + b2 = 169 Remember how we solve equations. We need to isolate thevariable by using the inverse operation. Subtract 25 from each side. b2 = 144b Find the principal square root of each side. b = 72 28. Find the Length of a Leg Dry ErasePractice What is the side length a in the triangle?12a15 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 TrianglesDry 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