Transformation of shapes
What is Transformation of shapes?
A rigid transformation that moves every point of a figure by the same distance in the same direction.
Key formula / rule: Translation
Key points
- Understand the definitions of translation, rotation, reflection, and scaling.
- Apply transformation rules to geometric shapes.
- Represent transformations using matrices.
- Perform combined transformations.
Common exam trap
Confusing clockwise and counter-clockwise rotation.
Definitions
- Term
Translation
- Meaning
A rigid transformation that moves every point of a figure by the same distance in the same direction.
- Term
Rotation
- Meaning
A transformation that turns a figure about a fixed point (center of rotation) by a specified angle.
- Term
Reflection
- Meaning
A transformation that creates a mirror image of a figure across a line or plane (axis of reflection).
- Term
Scaling
- Meaning
A transformation that changes the size of a figure by a scale factor, relative to a fixed point.
Learning objectives
Understand the definitions of translation, rotation, reflection, and scaling.
Apply transformation rules to geometric shapes.
Represent transformations using matrices.
Perform combined transformations.
Analyze the effect of transformations on geometric properties.
Formulae
- Name
Translation
- Note
Shifts point (x, y) by vector (tx, ty).
- Expression
x' = x + tx, y' = y + ty
- Name
Rotation about Origin (Counter-clockwise)
- Note
Rotates point (x, y) by angle θ.
- Expression
x' = x cos θ - y sin θ, y' = x sin θ + y cos θ
- Name
Reflection across X-axis
- Note
Mirrors point (x, y) across the x-axis.
- Expression
x' = x, y' = -y
- Name
Reflection across Y-axis
- Note
Mirrors point (x, y) across the y-axis.
- Expression
x' = -x, y' = y
- Name
Scaling from Origin
- Note
Scales point (x, y) by factors sx and sy.
- Expression
x' = sx * x, y' = sy * y
Prerequisites
Basic geometry (points, lines, angles).
Coordinate geometry (Cartesian system).
Basic matrix operations (multiplication, addition).
Trigonometry (sine, cosine).
Common mistakes
Confusing clockwise and counter-clockwise rotation.
Incorrectly applying the center of rotation or scaling.
Errors in matrix multiplication for combined transformations.
Misinterpreting the axis of reflection.
Keywords
Transformation
Translation
Rotation
Reflection
Scaling
Matrix
Geometry
Coordinate System
Geometric Manipulation
Practice preview
A square ABCD with vertices A(0,0), B(2,0), C(2,2), D(0,2) is transformed into a square A'B'C'D' with vertices A'(4, 2), B'(6, 2), C'(6, 0), D'(4, 0). Which sequence of transformations could achieve this?…
hard
A square is moved 3 units to the right and 2 units up. Which transformation has occurred?…
easy
A triangle is rotated 90 degrees clockwise about its centroid. What characteristic of the triangle remains unchanged?…
easy