Coordinate proofs

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Coordinate proofs

  1. 1. ANALYTIC GEOMETRY6-1 Coordinate Proofs
  2. 2. 6-1 Coordinate Proofs • Objective: To prove theorems from Geometry by using coordinates. • Suppose we had to prove or investigate a theorem about a right triangle. Which orientation of the coordinate axes seems preferable to work with? a b cUsually, because the math is easier, Figure a is least desirable. Figure c, whileawkward in its orientation may still be preferable. Figure b is probably our first choice.
  3. 3. 6-1 Coordinate Proofs• What’s true of triangles is true of other shapes as well.• It’s easy to see that both the trapezoid and the parallelogram are easier to work with if aligned with the axes:
  4. 4. Example 1: Midpoint of Hypotenuse•
  5. 5. Example 1: Midpoint of Hypotenuse•
  6. 6. Example 1: Midpoint of Hypotenuse•
  7. 7. Example 2: Median of a Trapezoid•
  8. 8. Example 2: Median of a Trapezoid•
  9. 9. Example 2: Median of a Trapezoid•
  10. 10. Example 3: Altitudes of a Triangle•
  11. 11. Example 3: Altitudes of a Triangle•
  12. 12. Example 3: Altitudes of a Triangle•
  13. 13. Methods Used in Coordinate Proofs1. To prove line segments equal, use the distance formulato show that they have the same length.2. To prove non-vertical lines parallel, show that they havethe same slope.3. To prove lines perpendicular, show that the product oftheir slopes is -1.4. To prove that two line segments bisect each other, usethe midpoint formula to show that each segment has thesame midpoint.5. To show that lines are concurrent, show that theirequations have a common solution.
  14. 14. Homework: pages 218 - 219#1, 3, 5, 7, 9, 11.

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