Presentation from textbook information for use in class on Coordinate Proofs.

1. ANALYTIC GEOMETRY
6-1 Coordinate Proofs

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 c
Usually, because the math is easier, Figure a is least desirable. Figure c, while
awkward in its orientation may still be preferable. Figure b is probably our first choice.

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. Example 1: Midpoint of Hypotenuse
•

5. Example 1: Midpoint of Hypotenuse
•

6. Example 1: Midpoint of Hypotenuse
•

7. Example 2: Median of a Trapezoid
•

8. Example 2: Median of a Trapezoid
•

9. Example 2: Median of a Trapezoid
•

10. Example 3: Altitudes of a Triangle
•

11. Example 3: Altitudes of a Triangle
•

12. Example 3: Altitudes of a Triangle
•

13. Methods Used in Coordinate Proofs
1. To prove line segments equal, use the distance formula
to show that they have the same length.
2. To prove non-vertical lines parallel, show that they have
the same slope.
3. To prove lines perpendicular, show that the product of
their slopes is -1.
4. To prove that two line segments bisect each other, use
the midpoint formula to show that each segment has the
same midpoint.
5. To show that lines are concurrent, show that their
equations have a common solution.

14. Homework: pages 218 - 219
#1, 3, 5, 7, 9, 11.