Mastering Geometric Proofs: How To Use Coordinate System To Prove Theorems

Coordinate proof Chapter 5 (p. 274)

A style of proof that involves placing geometric figures in a coordinate plane

A coordinate proof is a method of proving geometric theorems using algebra and the Cartesian coordinate system. In such a proof, geometric figures are represented by algebraic expressions and equations.

The steps involved in a coordinate proof are as follows:

1. Identify the geometric figure that needs to be proven.
2. Assign coordinates to the vertices or other important points of the figure.
3. Use the formulae for distance, midpoint, slope, and other properties of lines and angles to derive the equations that describe the figure.
4. Use algebra and logical reasoning to verify that the equations are true and that they describe the geometric figure as intended.
5. State the theorem that has been proven, in terms of the original geometric entities.

For example, consider the theorem that the diagonals of a rectangle are congruent. To prove this using a coordinate proof, we can assign coordinates to the four vertices of the rectangle, such as (0, 0), (0, a), (b, a), and (b,0), where a and b are the lengths of the sides of the rectangle. Then, we can use the distance formula to calculate the lengths of the two diagonals:

d1^2 = (b – 0)^2 + (a – 0)^2 = b^2 + a^2
d2^2 = (b – 0)^2 + (0 – a)^2 = b^2 + a^2

Since both lengths are equal, we can conclude that the diagonals are congruent.

Coordinate proofs can be useful in situations where traditional approaches such as Euclidean geometry or trigonometry are less effective or less applicable. They can also be helpful in visualizing geometric relationships and in developing intuition about algebraic representations of geometric concepts.

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