Questions: Coordinate Geometry Proofs

Question 1

Two line segments in a coordinate proof have slopes 3/4 and −4/3. What does this tell you about the lines?

• A. They are parallel
• B. They are perpendicular
• C. They are congruent
• D. They bisect each other

Answer: (B)

Two lines are perpendicular when their slopes are negative reciprocals — that is, m₁ × m₂ = −1. Here (3/4) × (−4/3) = −1, confirming perpendicularity. A common error is confusing this with just taking the reciprocal (3/4 and 4/3), which does not give perpendicular lines.

Question 2

To prove that two sides of a triangle are congruent in a coordinate geometry proof, it is sufficient to show that those sides have equal slopes.

• T. True
• F. False

Answer: False

Slope measures direction (steepness), not length. Two segments can share the same slope and yet be completely different lengths. Congruence requires using the distance formula d = √((x₂−x₁)² + (y₂−y₁)²). Equal slope proves parallelism; equal distance proves congruence.

Question 3

Why do coordinate geometry proofs often place one vertex at the origin and one side along the x-axis?

Model answer: To simplify the algebra — placing a vertex at (0, 0) and aligning a side with the x-axis sets several coordinates to zero, reducing the number of variables in distance and slope computations without changing any geometric properties.

Geometric properties (congruence, parallelism, bisection) are preserved under translation and rotation, so strategic placement is always valid. For example, a square with one vertex at the origin and one side along the x-axis has vertices (0,0), (a,0), (a,a), (0,a) — far cleaner than four arbitrary coordinates — making every slope, distance, and midpoint calculation simpler.