Translate P(-4,-5) by ⟨5,-1⟩.
Answer: (1,-6)
Reasoning: Add 5 to x and -1 to y.
Reasoning: Add 5 to x and -1 to y.
Unit 02 · Lesson 03 · Transformations, Symmetry & Congruence
Rotations About the Origin and Other Centers · Teacher Key
Construct and compare circumcenter, incenter, centroid, and orthocenter locations.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to construct and compare circumcenter, incenter, centroid, and orthocenter locations.
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
Represent motion with coordinate rules and use preserved distances and angles to reason about congruence and symmetry.
T_{\langle a,b\rangle}(x,y)=(x+a,y+b)R_{90^\circ}(x,y)=(-y,x)r_{y=x}(x,y)=(y,x)
Worked reasoning
Process: Apply the coordinate rule to each vertex, list the image coordinates, then verify preserved lengths or angles.
- 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 transformation rule must be applied to every vertex in the same order; changing only part of a figure does not create its image.
Khaemenes Academy · Open-Age Geometry
Rotations About the Origin and Other Centers
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Reflect Q(-4,-5) across the x-axis.
Answer: (-4,5)
Reasoning: Reflection across the x-axis changes the sign of y.
Reasoning: Reflection across the x-axis changes the sign of y.
Rotate R(-4,-5) 90° counterclockwise about the origin.
Answer: (5,-4)
Reasoning: Use (x,y)→(−y,x).
Reasoning: Use (x,y)→(−y,x).
Which measurements are preserved by every rigid motion?
Answer: Distances and angle measures
Reasoning: Translations, reflections, and rotations are isometries.
Reasoning: Translations, reflections, and rotations are isometries.
What is the order of rotational symmetry of a regular hexagon?
Answer: 6
Reasoning: It maps to itself six times in one full turn.
Reasoning: It maps to itself six times in one full turn.
Why does the order of transformations matter in a composition?
Answer: Different orders can produce different images
Reasoning: Transformation composition is generally not commutative.
Reasoning: Transformation composition is generally not commutative.