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The. Pythagorean. Theorem. 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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  1. The Pythagorean Theorem 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. 7.2 The Converse of the Pythagorean Theorem B • If c2 = a2 + b2, then ∆ABC is a right triangle. c a C A b

  40. Verify a Right Triangle C 16 12 • Is ∆ABC a right triangle? • Yes, it is a right triangle. A B 20

  41. Classifying Triangles C b a B A c B c a A C b B c a A C b

  42. Acute Triangles 5 4 • Show that the triangle is an acute triangle. • Because c2 < a2 + b2, the triangle is acute.

  43. Obtuse Triangles 12 8 • Show that the triangle is an obtuse triangle. • Because c2 > a2 + b2, the triangle is obtuse.

  44. Classify Triangles 6 5 • Classify the triangle as acute, right, or obtuse. • Because c2 > a2 + b2, the triangle is obtuse.

  45. Classify Triangles • Classify the triangle with the given side lengths as acute, right, or obtuse. • A. 4, 6, 7 • Because c2 < a2 + b2, the triangle is acute.

  46. Classify Triangles • Classify the triangle with the given side lengths as acute, right, or obtuse. • B. 12, 35, 37 • Because c2 = a2 + b2, the triangle is right.

  47. 7.3 Similar Right Triangles Geometry Mr. Lopiccolo

  48. Objectives/Assignment • Solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle. • Use a geometric mean to solve problems such as estimating a climbing distance. • Assignment: pp. 453-454 (4-26) even

  49. In Lesson 6.4, you learned that two triangles are similar if two of their corresponding angles are congruent. For example ∆PQR ~ ∆STU. Recall that the corresponding side lengths of similar triangles are in proportion. Proportions in right triangles

  50. Cut an index card along one of its diagonals. On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles. You should now have three right triangles. Compare the triangles. What special property do they share? Explain. Tape your group’s triangles to a piece of paper and place in labwork. Activity: Investigating similar right triangles. Do in pairs or threes

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