Point A is at 7 and point B is at 19 on a number line. Find AB.
Answer: 12
Reasoning: Distance is |19−7|=12.
Reasoning: Distance is |19−7|=12.
Unit 01 · Lesson 03 · Foundations, Constructions & Geometric Reasoning
Distance, Midpoint & Segment Partition · Teacher Key
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
Use definitions, measurements, constructions, and logical statements to establish a precise geometric argument.
AB=|b-a|M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)m\angle A+m\angle B=180^\circ
Worked reasoning
Process: State what is given, translate each fact into a definition or theorem, and justify every conclusion in order.
- 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: A diagram suggests relationships, but only stated markings, definitions, and proven results may be used as evidence.
Khaemenes Academy · Open-Age Geometry
Distance, Midpoint & Segment Partition
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Find the midpoint of A(0,1) and B(7,4).
Answer: (3.5,2.5)
Reasoning: Average the x-coordinates and the y-coordinates separately.
Reasoning: Average the x-coordinates and the y-coordinates separately.
Point M lies between A and B. If AM=11 and MB=35, find AB.
Answer: 46
Reasoning: Use segment addition: AB=AM+MB.
Reasoning: Use segment addition: AB=AM+MB.
Write the correct notation for the ray that starts at A and passes through B.
Answer: ray AB
Reasoning: The endpoint letter is written first.
Reasoning: The endpoint letter is written first.
Explain why the length of segment AB is written without a bar, while the segment itself is written with a bar.
Answer: AB is a number; segment AB is a geometric set
Reasoning: Notation distinguishes the object from its measure.
Reasoning: Notation distinguishes the object from its measure.
A point partitions a 30-unit segment in the ratio 2:3. Find the two part lengths.
Answer: 12 and 18
Reasoning: There are 5 ratio parts; each is 6 units.
Reasoning: There are 5 ratio parts; each is 6 units.