Proofs Using Coordinate Geometry

How Does a Coordinate Proof Work?

Proofs using coordinate geometry use the slope, midpoint, and distance formulas to proof rules and theorems.

Ex: Prove a Rectangle Has Congruent Diagonals

Step 1: Place the figure on the xy-axis

Step 2: Correctly label the pointsStep 3: Write a Given and Prove

statementStep 4: Use slope, mp, or distance

formulasStep 5: Write a concluding

statement

( a , b )

( 0 , 0 )

( 0 , b )

D

( a , 0 )

CB

A

Given: ABCD is a rectangleProve: Diagonals are = (AC=BD)

2 2 2 2

2 2 2 2

0 ( 0)

( 0) (0 )

AC a b a b

BD a b a b

AC and BD have the same length. Therefore the diagonals of rectangles are congruent.

What types of proofs can be done with C.G.?

The slope formula can show: Segments are parallel. Segments are perpendicular. A figure has right angles.

The distance formula can show: Segments have the same length Two segments bisect each other

The midpoint formula can show: The location of a midpoint Two segments bisect each other.

Deciding whether C.G. will work on a Proof.

State whether each of the following can be determined with coordinate geometry. EF=GH

Yes, with the distance formula BD ll AC

Yes, with the slope formula <A=<B

No, unless both are right angles FG bisects JG

Yes, with the distance or midpoint formulas

Deciding whether C.G. will work on a Proof.

State whether each of the following can be determined with coordinate geometry. Triangle LMN is isosceles

Yes, with the distance formula The diagonals of Kite QRST are perpendicular

Yes, with the slope formula <C and <D are supplementary

No.

Homework

P 335 (12-24)Worksheet - Proofs Using the Coordinate

Plane