6 1 coordinate proofs
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Analytic geometry uses algebraic methods to study and prove geometric problems using coordinates. This chapter discusses using coordinate proofs to prove theorems from geometry. It provides instructions on how to construct a coordinate proof by drawing and labeling a diagram, listing given information, stating what will be proved, using algebra to add to the diagram and prove the statement, and writing a conclusion. Common algebraic methods used in proofs include the distance formula, identifying parallel and perpendicular lines, and finding midpoints. Examples provided are proving properties of right triangles, trapezoids, and triangles.
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6 1 coordinate proofs
- 1. Chapter 6 – Analytic Geometry 6-1 Coordinate Proofs Objectives: 1. To prove theorems from geometry by using coordinates.
- 2. What is Analytic Geometry? the study of geometric problems using algebraic methods. For example: ◦ Distance Formula ◦ Midpoint Formula
- 3. Placing Coordinate Axes For example: For a right triangle, axes should be placed so the legs lie on them
- 4. Parallelograms/trapezoids often want a parallel side on the x-axis and a vertex at the origin
- 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. 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. Example 1: Prove that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
- 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. Example 3: Prove that the altitudes of a triangle are concurrent (meet at one point).