Coordinate Proofs with Slope
A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.
Coordinate proofs use algebraic evidence to prove geometric facts. Slopes show whether sides are parallel or perpendicular, and distances show whether sides are congruent.
What you'll learn
- Use slope for parallel sides
- Use slope for perpendicular sides
- Classify coordinate figures
Worked example
Problem. A quadrilateral has opposite sides with slopes 2 and 2. What does that show?
- Lines with equal slopes are parallel if they are distinct.
- The opposite sides have the same slope.
- So those sides are parallel.
Answer: Those opposite sides are parallel.
Practice problems
1. Lines with slopes 3 and 3 are...
Choices: Parallel · Perpendicular · Neither · Vertical
Show solution
- Warm-up: First identify exactly what the question is asking: Lines with slopes 3 and 3 are...
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Equal slopes are parallel.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Parallel
2. Lines with slopes 2 and -1/2 are...
Choices: Perpendicular · Parallel · Neither · Horizontal
Show solution
- Warm-up: First identify exactly what the question is asking: Lines with slopes 2 and -1/2 are...
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- They are opposite reciprocals.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Perpendicular
3. What slope is perpendicular to 4?
Show solution
- Core Practice: First identify exactly what the question is asking: What slope is perpendicular to 4?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Use the opposite reciprocal.
- Check the result by substituting or estimating: the response should match -1/4 and make sense in the original problem.
Answer: -1/4
4. A quadrilateral has both pairs of opposite sides parallel. It is at least a...
Choices: Parallelogram · Triangle · Circle · Kite only
Show solution
- Core Practice: First identify exactly what the question is asking: A quadrilateral has both pairs of opposite sides parallel. It is at least a...
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- This is the definition of a parallelogram.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Parallelogram
5. A quadrilateral has four congruent sides. It is a...
Choices: Rhombus · Rectangle always · Circle · Triangle
Show solution
- Challenge: First identify exactly what the question is asking: A quadrilateral has four congruent sides. It is a...
- For congruence, match corresponding sides and angles in the same order before choosing a theorem or equation.
- A rhombus has four congruent sides.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Rhombus
Practice this interactively with instant feedback and an AI tutor.
Practice Coordinate Proofs with Slope Take the free placement check