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Pythagoras Theorem

Pythagoras Theorem. Book 2 Chapter 6. c. a. b. This is a right triangle:. We call it a right triangle because it contains a right angle. The measure of a right angle is 90 o. 90 o. in the. The little square. angle tells you it is a. right angle. 90 o.

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Pythagoras Theorem

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  1. Pythagoras Theorem Book 2 Chapter 6 c a b

  2. This is a right triangle:

  3. We call it a right triangle because it contains a right angle.

  4. The measure of a right angle is 90o 90o

  5. in the The little square angle tells you it is a right angle. 90o

  6. About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.

  7. 5 3 4 Pythagorus realized that if you have a right triangle,

  8. 5 3 4 and you square the lengths of the two sides that make up the right angle,

  9. 5 3 4 and add them together,

  10. 5 3 4 you get the same number you would get by squaring the other side.

  11. Is that correct? ? ?

  12. 10 8 6 It is. And it is true for any right triangle.

  13. The two sides which come together in a right angle are called

  14. The two sides which come together in a right angle are called

  15. The two sides which come together in a right angle are called legs.

  16. The lengths of the legs are usually called a and b. a b

  17. The side across from the right angle is called the hypotenuse. a b

  18. And the length of the hypotenuse is usually labeled c. c a b

  19. The relationship Pythagorus discovered is now called The Pythagorean Theorem: c a b

  20. The Pythagorean Theorem says, given the right triangle with legs a and b and hypotenuse c, c a b

  21. then c a b

  22. You can use The Pythagorean Theorem to solve many kinds of problems. Suppose you drive directly west for 48 miles, 48

  23. Then turn south and drive for 36 miles. 48 36

  24. How far are you from where you started? 48 36 ?

  25. Using The Pythagorean Theorem, 48 482 + 362 = c2 36 c

  26. 48 482 + 362 = c2 36 c Why? Can you see that we have a right triangle?

  27. 48 482 + 362 = c2 36 c Which side is the hypotenuse? Which sides are the legs?

  28. Then all we need to do is calculate:

  29. So, since c2 is 3600, c is 60. And you end up 60 miles from where you started. So, since c2 is 3600, c is 48 36 60

  30. 15" 8" Find the length of a diagonal of the rectangle: ?

  31. 15" 8" Find the length of a diagonal of the rectangle: ? c b = 8 a = 15

  32. c b = 8 a = 15

  33. 15" 8" Find the length of a diagonal of the rectangle: 17

  34. Practice using The Pythagorean Theorem to solve these right triangles:

  35. c 5 12 = 13

  36. b 10 26

  37. b 10 26 = 24 (a) (c)

  38. 12 b 15 = 9

  39. Support Beam: The skyscrapers are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.

  40. Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).

  41. Solution: (hypotenuse)2 = (leg)2 + (leg)2 x2 = (23.26)2 + (47.57)2 x2 = √ (23.26)2 + (47.57)2 x ≈ 13

  42. Ladder Problem • A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?

  43. Ladder ProblemSolution • First draw a diagram that shows the sides of the right triangle. • Label the sides: • Ladder is 25 m • Distance from house is 7 m • Use a2 + b2 = c2 to solve for the missing side. Distance from house: 7 meters

  44. Ladder ProblemSolution • 72 + b2 = 252 • 49 + b2 = 625 • b2 = 576 • b = 24 m

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