4.2.4 Introduction to Coordinate Proof Key Objectives • Position figures in the coordinate plane for use in coordinate proofs. • Prove geometric concepts by using coordinate proof. Key Terms • A coordinate proof is a style of proof that uses coordinate geometry and algebra. Example 1 Positioning a Figure in the Coordinate Plane
The figure in a coordinate proof is either given or placed on a coordinate plane. If coordinates of the figure’s vertices are not given in the conjecture, then the figure can be placed anywhere on the coordinate plane. However, there are some guidelines for positioning a figure that will result in easier steps in the proof.
In this example, a figure is given. The figure is a rectangle with length 6 units and width 4 units. The rectangle can be positioned (placed) on the coordinate plane in any of the four quadrants. However, it is typically easiest to place a figure in the first quadrant so that the coordinates in each vertex’s ordered pair will be a positive number. The rectangle could be oriented in any way, but placing the rectangle so that its sides are vertical and horizontal, ideally with one side on the x -axis and one side on the y -axis, is best. So, Position 1 is the best position for this rectangle because the coordinates of the vertices are all positive whole numbers and two of the rectangle’s sides are placed along an axis.
Example 2 Writing a Proof Using Coordinate Geometry
Make a plan for proving that the area of △ ABC is twice the area of △ DBC . Find the area of each triangle and then multiply the area of △ DBC by 2 to determine whether the area of △ ABC = 2 (the area of △ DBC ). Remember, the area of a triangle is 1/2 the product of the triangle’s base and height (where the height is the perpendicular distance from the base to the opposite vertex). So, the area of △ ABC is (1/2)(5) (8) = 20 units 2 . Before the area of △ DBC can be found, the location of D must be determined. Since D is given to be the midpoint between A and C , the location of D can be found by using the midpoint formula. The y -coordinate of D is the height of △ DBC . Now the area of △ DBC can be found.
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