Preparing Proofs in Coordinate Geometry
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     In the seventeenth century, the French mathematician Rene Descartes applied algebraic principles to geometric situations.  This blending of algebra and geometry is referred to as analytic geometry.  Because this process often involves placing geometric figures in a coordinate plane, it is also more commonly known as coordinate geometry.

  Coordinate geometry proofs employ the use of formulas such as the Distance Formula, the Slope Formula and/or the Midpoint Formula as well as postulates, theorems and definitions.

Distance Formula

Slope Formula
Midpoint Formula

 

When developing a coordinate geometry proof:

1.  draw and label the graph
2.  state the formulas you will be using
3.  show ALL work (if you are using your graphing
      calculator, be sure to show your screen displays as
      part of your work.)
4.  have a concluding sentence stating
    what you have proven and why it is true.


Example 1: 

Given: 
Prove:  is isosceles

Read the question carefully.  The word isosceles, by definition, tells you that you are looking for two congruent sides.  Since congruent implies "of equal length" and the word length implies "distance", you will use the Distance Formula.

Draw a neat, labeled graph for the problem.

State the formula that you will be using.








Show ALL work!!  Since we are looking for two sides of equal length, you can STOP when you find the two sides.  Look at the figure before you begin and choose the two sides that you "think" are of equal length. 

The triangle is isosceles because it has two congruent sides.

End with a concluding sentence stating WHY you know the triangle is isosceles.


Example 2:

Read the question carefully.  The word trapezoid, by definition, tells you that you are looking for a figure with ONLY ONE set of parallel sides.  Lines are parallel when they have the same slope.  You will use the Slope Formula.

Draw a neat, labeled graph for the problem.

State the formula that you will be using.

Show ALL work!!  We are looking for ONE set of parallel sides AND one set on non-parallel sides.

Be sure to state the connection between the slopes and the sides being parallel or non-parallel.

End with a concluding sentence stating WHY you know the figure is a trapezoid.