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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Apply three rigid motions.
- Record intermediate coordinates.
- Compare transformation order.
Before you start
Signed coordinates and clockwise/counterclockwise directions.
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
A design point starts at P = (2, 1). Reflect it across the y-axis, rotate it 90° counterclockwise about the origin, then translate by (3, 4). Where does it finish?
Why this math matters
Graphic layouts and geometric constructions use sequences of transformations. Each instruction changes the input to the next, so remembering the order matters as much as knowing each coordinate rule.
Set up the model
A useful answer starts with clear assumptions:
- Every rotation is about the origin.
- The reflection line is the y-axis.
- The same sequence applies to every vertex of the design.
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.
Move a design with reflections, rotations, and translations
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A design point starts at P = (2, 1). Reflect it across the y-axis, rotate it 90° counterclockwise about the origin, then translate by (3, 4). Where does it finish?
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.
Reflect first
(x, y) → (−x, y); (2, 1) → (−2, 1)
A mirror across the vertical axis changes the horizontal sign and preserves the vertical coordinate.
Turn the reflected point
(x, y) → (−y, x); (−2, 1) → (−1, −2)
Use the reflected point as input. The rotation rule swaps coordinate roles and negates the former vertical value.
Translate last
(−1, −2) + (3, 4) = (2, 2)
Add the displacement component by component. Translation moves all points equally and preserves their separations.
The result
The point finishes at (2, 2).
All three operations preserve lengths and angle sizes, so the final design is congruent to its original. They can change orientation and location without changing size.
Common mistakes to catch
- Reflecting across the y-axis negates x, not y.
- Applying the rotation to the original point skips the reflection.
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
Rotate (3, −2) 90° counterclockwise about the origin.
Show a hint
Apply (x, y) → (−y, x).
Reveal answer and explanation
(2, 3)
Negate −2 for the new x and use 3 for the new y.
Practice 2
Translate (1, 0) by (2, 0), then rotate 90° counterclockwise. What if you reverse these operations?
Show a hint
Keep both intermediate points.
Reveal answer and explanation
(0, 3) first; (2, 1) in reverse order
The first route is (3, 0) → (0, 3). The reverse is (0, 1) → (2, 1).
Take the idea with you
When combining design operations, write the intermediate state after every operation rather than relying on a mental shortcut.
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: Read an ellipse from its equation and foci
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