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

5. Example:
 Find
the missing
coordinates:

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).