Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to establish congruence with a sequence of motions.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Quarter-turns and translations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Triangle A has vertices (0,0), (2,0), (0,1). Triangle B has vertices (5,3), (5,5), (4,3). Give a rotation followed by a translation that maps A onto B.
Why this math matters
Congruent figures can be matched by rigid motions that preserve every distance and angle. Use a sequence of allowable transformations to justify a geometric match instead of relying on the sketch.
Set up the model
A useful answer starts with clear assumptions:
- The listed vertex order supplies the intended correspondence.
- All points use the same coordinate scale.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Establish congruence with a sequence of motions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Triangle A has vertices (0,0), (2,0), (0,1). Triangle B has vertices (5,3), (5,5), (4,3). Give a rotation followed by a translation that maps A onto B.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the quantities
Rotate A 90° counterclockwise: (0,0), (0,2), (−1,0)
Use (x,y) → (−y,x) for each vertex.
Apply the relationship
Translate by (5,3): (5,3), (5,5), (4,3)
The same translation aligns all three rotated vertices with B.
Check and interpret
A and B are congruent
A rotation and a translation preserve distances and angles, and the constructed images coincide.
The result
A and B are congruent
A rotation and a translation preserve distances and angles, and the constructed images coincide.
Common mistakes to catch
- A figure may be congruent even if it faces a different direction.
- Matching appearance alone is weaker than constructing a motion that aligns corresponding points.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Does reflecting a triangle change its congruence class?
Show a hint
A reflection preserves distances and angle measures.
Reveal answer and explanation
No; its image is congruent
Orientation reverses, but shape and size remain the same.
Practice 2
Does dilating a nondegenerate triangle by factor 2 produce a congruent image?
Show a hint
Compare a corresponding nonzero side length.
Reveal answer and explanation
No; it produces a similar image
Every side doubles, so size changes even though angles stay equal.
Take the idea with you
Use a sequence of allowable transformations to justify a geometric match instead of relying on the sketch.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Explain constant slope with similar triangles
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