Coordinate Proof

A coordinate proof is a method used to prove geometric statements by placing geometric figures in a coordinate plane and assigning variables to the coordinates of points. These variables help prove geometric statements algebraically. Consider the following statement.

There are three steps to follow to prove this statement using a coordinate proof.

1

Position the Figure in a Coordinate Plane

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The first step is to place the figure in a coordinate plane. Several strategies can be applied to simplify the next steps in the process.

• Use the origin as a vertex or the center of the figure.
• Keep the figure in Quadrant I, if possible.
• Use the axes as sides of the figure.

Considering these strategies, place a rectangle in a coordinate plane and label the vertices as A, B, C, and D.

2

Assign Coordinates to Each Vertex

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Since point A is on the origin, its coordinates are (0,0). Because B lies on the x-axis, its y-coordinate is 0. In a similar vein, since D lies on the y-axis, its x-coordinate is 0. Assign variables a and b for the x-coordinate of B and y-coordinate of D, respectively.

In this step it is important to assign as few variables as possible. Since the figure is a rectangle, each of its interior angles is 90^(∘). This implies that the coordinates of C are ( a, b).

3

Write a Coordinate Proof

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Draw the diagonals of the rectangle.

Calculate the length of the diagonals AC and DB using the Distance Formula.

d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Segment	Endpoints	Substitute	Simplify
AC	A( 0,0) and C( a,b)	AC=sqrt(( a - 0)^2+( b- 0)^2)	AC=sqrt(a^2+b^2)
DB	D( 0,b) and B( a,0)	DB=sqrt(( a- 0)^2+( 0- b)^2)	DB=sqrt(a^2+b^2)

Both segments have the same length. Therefore, by the definition of congruent segments, the diagonals are congruent. AC≅ DB