Describe the locus of points equidistant from two fixed points A and B.
Answer: The perpendicular bisector of AB
Reasoning: Every point on it has equal distance to A and B.
Reasoning: Every point on it has equal distance to A and B.
Unit 09 · Lesson 07 · Analytic Geometry, Loci & Conic Connections
Voronoi Thinking, Nearest-Point Regions & Design · Teacher Key
Partition a region by nearest-site distance and explain boundaries as perpendicular-bisector loci.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to partition a region by nearest-site distance and explain boundaries as perpendicular-bisector loci.
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
Describe loci and intersections as solution sets defined by distance, coordinate, and boundary conditions.
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}(x-h)^2=4p(y-k)
Worked reasoning
Process: Translate the verbal distance condition into an equation or construction, solve or graph it, and test representative points.
- 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 locus is the complete set of all points satisfying a condition, not just several sample points that happen to work.
Khaemenes Academy · Open-Age Geometry
Voronoi Thinking, Nearest-Point Regions & Design
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Describe the locus of points equidistant from two intersecting lines.
Answer: The two angle bisectors
Reasoning: Both internal and external angle bisectors satisfy the condition.
Reasoning: Both internal and external angle bisectors satisfy the condition.
Find the distance from (3,4) to the x-axis.
Answer: 4
Reasoning: Distance to the x-axis is |y|.
Reasoning: Distance to the x-axis is |y|.
What condition makes a line tangent to a circle?
Answer: It intersects the circle at exactly one point
Reasoning: Equivalently, the center-to-line distance equals the radius.
Reasoning: Equivalently, the center-to-line distance equals the radius.
Define a parabola geometrically.
Answer: Points equidistant from a focus and a directrix
Reasoning: This distance condition generates the conic.
Reasoning: This distance condition generates the conic.
Why are Voronoi boundaries portions of perpendicular bisectors?
Answer: Boundary points are equidistant from two competing sites
Reasoning: Nearest-site regions change where distances tie.
Reasoning: Nearest-site regions change where distances tie.