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# 6 1 coordinate proofs

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### 6 1 coordinate proofs

1. 1. Chapter 6 – Analytic Geometry 6-1 Coordinate Proofs Objectives: 1. To prove theorems from geometry by using coordinates.
2. 2. What is Analytic Geometry?  the study of geometric problems using algebraic methods.  For example: ◦ Distance Formula ◦ Midpoint Formula
3. 3. Placing Coordinate Axes  For example:  For a right triangle, axes should be placed so the legs lie on them
4. 4.  Parallelograms/trapezoids often want a parallel side on the x-axis and a vertex at the origin
5. 5. Example:  Find the missing coordinates:
6. 6. How to Construct a Coordinate Proof:  Draw and label a coordinate diagram  List given information  State what you will prove  Use given info to add to the diagram  Use algebra to prove statement  Write conclusion: ◦ “Therefore, blah = blah.”
7. 7. Common Methods to use:  To prove:  Segments are equal  use distance formula  Lines are parallel  show slopes are equal  Lines are perpendicular  show slopes multiply to -1  Segments bisect  show they have the same midpoint  Lines are concurrent  show equations have a common solution
8. 8. Example 1:  Prove that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
9. 9. Example 2:  Prove that the median of a trapezoid is parallel to the bases and has length equal to the average length of the bases.
10. 10. Example 3:  Prove that the altitudes of a triangle are concurrent (meet at one point).