Reflection
The mirror line is invariant point-by-point.
🎯 Learning Objectives
- define invariant points and invariant lines;
- distinguish setwise and pointwise invariance;
- identify invariant features from matrices and diagrams;
- use invariance to check transformations.
Introduction
This lesson develops Invariant Points and Lines from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Invariant Point
An invariant point maps to itself. For a rotation, the centre is invariant. For an enlargement, the centre is invariant.
Invariant Line
A line is invariant if it maps onto itself. It may be fixed point-by-point or only as a whole set.
Horizontal shear
The x-axis is invariant point-by-point.
Stretch in x-direction
The y-axis is invariant point-by-point.
Rotation
Usually only the centre is invariant.
Practice Questions
- State an invariant point under rotation.
- State the invariant line under reflection in y=3.
- What is invariant under enlargement about C?
- Why are invariant features useful?
✅ Check Your Work
Show answers and working
- The centre of rotation.
- The line y=3.
- The centre C.
- They help identify the transformation and check whether the image is correct.
🧠 Learn It Visually: Invariant Points and Lines
Invariant means unchanged by the transformation. A point may stay fixed, or an entire line may map onto itself.
Invariant point
A point P is invariant when P′=P.
Invariant line pointwise
Every point on the line stays exactly where it is.
Invariant line as a set
Points may move along the line, but the line still maps onto itself.
🎯 Identify the Transformation
Study the object and image, then choose the most complete description.
Look at size, orientation and corresponding movements.
Think Like a Mathematician: Can a line be invariant even if individual points on it move?
🧷 Remember
Always clarify whether a line is fixed point-by-point or merely maps onto itself.
⚠️ Common Mistake
Saying an axis is invariant without checking the actual transformation.
🎯 Exam Tip
Substitute a general point on the proposed line into the coordinate rule and verify that its image remains on the line.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.