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 05 · Analytic Geometry, Loci & Conic Connections
Circle-Line Intersections and Tangency · Teacher Key
Describe loci and intersections as solution sets defined by distance, coordinate, and boundary conditions.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to describe loci and intersections as solution sets defined by distance, coordinate, and boundary conditions.
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
Circle-Line Intersections and Tangency
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.