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Distance Formula and Pythagorean Theorem - Problem of the Day Lesson

This lesson covers the use of the Distance Formula and Pythagorean Theorem to find distances and identify right triangles. Includes quizzes.

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Distance Formula and Pythagorean Theorem - Problem of the Day Lesson

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  1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

  2. Warm Up Find the distance between each pair of points. 1. (8, 2) and (8, 7)‏ 2. (–2, 4) and (5, 4)‏ 3. (–1, –1) and (9, –1)‏ 4. (–8, –4) and (–2, –4)‏ 5 units 7 units 10 units 6 units

  3. Problem of the Day The sum of the squares of two positive numbers is 100. One number is two more than the other. What are the numbers? 6 and 8

  4. Learn to use the Distance Formula and the Pythagorean Theorem and its converse to solve problems.

  5. 58 = c Additional Example 1: Marketing Application What is the diagonal length of the projector screen? Use the Pythagorean Theorem 72 + 32 = c2 Simplify. 49 + 9 = c2 58 = c2 Add. 7.615c The diagonal length should be given as about 7.62 feet.

  6. 12 ft 8 ft 208 = c Check It Out: Example 1 What is the diagonal length of the projector screen? Use the Pythagorean Theorem 122 + 82 = c2 Simplify. 144 + 64 = c2 208 = c2 Add. 14.42c The diagonal length should be given as about 14.4 feet.

  7. Additional Example 2A: Finding Distance on the Coordinate Plane Find the distances between the points to the nearest tenth. J and K Let A be (x1, y1)² and B be (x2, y2)². Use the Distance Formula.

  8. d = (0 – (–4))² + (–3 – 0)² d = (4)² + (–3)² d = 16 + 9 Additional Example 2A Continued Find the distances between the points to the nearest tenth. Substitute. Subtract. Simplify powers. The distance between A and B is 5 units.

  9. Additional Example 2B: Finding Distance on the Coordinate Plane Find the distances between the points to the nearest tenth. L and M Let A be (x1, y1)² and B be (x2, y2)². Use the Distance Formula.

  10. d = (5– 4)² + (–3 – 0)² d = (1)² + (–3)² d = 1 + 9 Additional Example 2B Continued Find the distances between the points to the nearest tenth. Substitute. Subtract. Simplify powers. The distance between L and M is about 3.2 units.

  11. Check It Out: Example 2A Find the distances between the points to the nearest tenth. J and L Let A be (x1, y1)² and B be (x2, y2)². Use the Distance Formula.

  12. d = (4– (–4))² + (0 – 0)² d = (8)² + (0)² d = 64 + 0 Check It Out: Example 2A Continued Find the distances between the points to the nearest tenth. Substitute. Subtract. Simplify powers. The distance between J and L is 8 units.

  13. Check It Out: Example 2B Find the distances between the points to the nearest tenth. K and M Let A be (x1, y1)² and B be (x2, y2)². Use the Distance Formula.

  14. d = ((–4) – 0)² + (– 3 – (–3)² d = (–4)² + (0)² d = 16 + 0 Check It Out: Example 2B Continued Find the distances between the points to the nearest tenth. Substitute. Subtract. Simplify powers. The distance between K and M is 4 units.

  15. Additional Example 3A: Identifying a Right Triangle Tell whether the given side lengths form a right triangle. 9, 12, 15 a2 + b2 = c2 Compare a² + b² to c². 92 + 122 = 152 Substitute. 81 + 144 = 225 Simplify. Add. 225 = 225 √ The side lengths form a right triangle.

  16. Additional Example 3B: Identifying a Right Triangle Tell whether the given side lengths form a right triangle. 8, 10, 13 a2 + b2 = c2 Compare a² + b² to c². 82 + 102 = 132 Substitute. 63 + 100 = 169 Simplify. Add. 163 ≠ 169 x The side lengths do not form a right triangle.

  17. Check It Out: Example 3A Tell whether the given side lengths form a right triangle. 5, 6, 9 a2 + b2 = c2 Compare a² + b² to c². 52 + 62 = 92 Substitute. 25 + 36 = 81 Simplify. Add. 61 ≠ 81 x The side lengths do not form a right triangle.

  18. Check It Out: Example 3B Tell whether the given side lengths form a right triangle. 8, 15, 17 a2 + b2 = c2 Compare a² + b² to c². 82 + 152 = 172 Substitute. 64 + 225 = 289 Simplify. Add. 289 = 289 √ The side lengths form a right triangle.

  19. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  20. Lesson Quiz: Part I Find the length of the diagonal of the rectangle. 1. width = 9 in., length = 40 in. 41 in. 2. base length = 20 m, height = 15 m. 25 m

  21. Lesson Quiz: Part II Find the distance between the points to the nearest tenth. 3.Q and R ≈ 6.3 4.S and T ≈ 2.2

  22. Lesson Quiz: Part III Tell whether the given side lengths form a right triangle. 5. 12 cm, 13 cm, 16 cm no 6. 11 ft, 60 ft, 61 ft yes

  23. Lesson Quiz for Student Response Systems 1. Find the length of the diagonal of the rectangle. width = 5 m, length = 12 m A. 8 m B. 11 mC.13 m D.17 m

  24. Lesson Quiz for Student Response Systems 2. Find the distance between the points to the nearest tenth. Q and S A. 6.0 m B. 6.1 mC.7.0 m D.37.1 m

  25. Lesson Quiz for Student Response Systems 3. Tell whether the given side lengths form a right triangle. 12 ft, 13 ft, 14 ft A. yes B. no

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