Find the slope through A(-4,0) and B(4,3).
Unit 04 · Lesson 03 · Parallel Lines, Slopes & Coordinate Proof
Slope as Direction and Rate
Combine angle relationships, slope, distance, midpoint, and equations of lines to prove geometric properties analytically.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to combine angle relationships, slope, distance, midpoint, and equations of lines to prove geometric properties analytically.
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
Slope as Direction and Rate
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Find the midpoint of A(-4,0) and B(4,3).
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).