Chapter 6 – Analytic Geometry

6-1 Coordinate ProofsObjectives:

1. To prove theorems from geometry by using coordinates.

What is Analytic Geometry?

the study of geometric problems using algebraic methods.

For example:◦Distance Formula

◦Midpoint Formula

Placing Coordinate AxesFor example:For a right triangle, axes should be

placed so the legs lie on them

Parallelograms/trapezoids often want a parallel side on the x-axis and a vertex at the origin

Example:Find the missing

coordinates:

How to Construct a Coordinate Proof:Draw and label a coordinate diagramList given informationState what you will proveUse given info to add to the diagramUse algebra to prove statementWrite conclusion:◦“Therefore, blah = blah.”

Common Methods to use:To prove:Segments are equal use distance

formulaLines are parallel show slopes are

equalLines are perpendicular show slopes

multiply to -1Segments bisect show they have the

same midpointLines are concurrent show equations

have a common solution

Example 1:Prove that the midpoint of the

hypotenuse of a right triangle is equidistant from the three vertices.

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.

Example 3:Prove that the altitudes of a

triangle are concurrent (meet at one point).