Find the slope through A(-5,-3) and B(7,8).
Unit 04 · Lesson 05 · Parallel Lines, Slopes & Coordinate Proof
Coordinate Proof with Distance, Midpoint & Slope
Partition a segment numerically and on the coordinate plane; distinguish distance from directed change.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to partition a segment numerically and on the coordinate plane; distinguish distance from directed change.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Combine angle relationships, slope, distance, midpoint, and equations of lines to prove geometric properties analytically.
m=\frac{y_2-y_1}{x_2-x_1}d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}m_1m_2=-1
Worked reasoning
Process: Compute only the measurements needed for the claim, compare them with a definition, and write a conclusion tied to the coordinates.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: Perpendicular slopes are negative reciprocals only when both slopes are defined; vertical and horizontal lines must be treated separately.
Khaemenes Academy · Open-Age Geometry
Coordinate Proof with Distance, Midpoint & Slope
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Find the midpoint of A(-5,-3) and B(7,8).
If one nonvertical line has slope 3/4, what is the slope of a perpendicular line?
What angle relationship is formed by alternate interior angles when lines are parallel?
What must be shown with slopes to prove a quadrilateral is a parallelogram?
Write an equation of the horizontal line through (4,−3).