Transformations
Explore how shapes move and change on the coordinate grid through translations, reflections, rotations, and dilations.
What Are Transformations?
A transformation changes the position, orientation, or size of a shape. The original shape is called the object and the result is called the image.
Translation
Sliding — same size, same orientation
Reflection
Flipping — mirror image
Rotation
Turning — about a centre point
Dilation
Enlarging or reducing — same shape
Translation and Reflection
Translation (Slide)
Every point moves the same distance in the same direction. Described as a vector, e.g. (3, −2) means right 3, down 2.
Reflection (Flip)
Every point is flipped over a mirror line. Each point in the image is the same distance from the mirror line as the original point.
Rotation and Dilation
Rotation (Turn)
Every point turns around a centre of rotation by a given angle (clockwise or anticlockwise). The shape stays the same size.
To describe a rotation, state:
- The angle (e.g. 90°, 180°)
- The direction (clockwise / anticlockwise)
- The centre of rotation (e.g. origin)
Dilation (Enlargement / Reduction)
Every point moves closer to or further from a centre of dilation by a scale factor. The shape stays the same but changes size.
Scale factor > 1: enlargement
Scale factor between 0 and 1: reduction
e.g. scale factor 2 doubles all lengths
Congruence and Similarity After Transformation
Congruent (Same Size & Shape)
Translations, reflections, and rotations produce congruent images — the size does not change, only the position/orientation.
Similar (Same Shape, Different Size)
Dilations produce similar images — the shape is preserved but the size changes according to the scale factor.
Key Vocabulary
Object & Image
The object is the original shape; the image is the result after transformation.
Mirror Line
The line of symmetry used in a reflection. Each image point is the same distance from this line as the object point.
Centre of Rotation
The fixed point around which a shape is rotated. Often the origin (0, 0) in Year 7.
Scale Factor
The number by which all lengths are multiplied in a dilation. e.g. scale factor 3 triples all side lengths.
Worked Examples
Translate the point A(2, 3) by the vector (4, −2).
Step 1: Add the x-component: 2 + 4 = 6
Step 2: Add the y-component: 3 + (−2) = 1
Answer: A'(6, 1)
Reflect B(3, −4) in the x-axis.
When reflecting in the x-axis, the x-coordinate stays the same and the y-coordinate changes sign.
Step 1: x stays: 3
Step 2: y changes sign: −4 → 4
Answer: B'(3, 4)
A triangle has vertices at (1,1), (3,1), (1,4). Dilate it with centre (0,0) and scale factor 2.
Multiply each coordinate by the scale factor of 2:
- (1,1) → (2,2)
- (3,1) → (6,2)
- (1,4) → (2,8)
The image is twice as large but has the same shape (similar).
Knowledge Check
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Key Concepts Summary
- ●A translation slides a shape by adding a vector to every coordinate.
- ●A reflection flips a shape over a mirror line; reflecting in the x-axis changes the y-sign, the y-axis changes the x-sign.
- ●A rotation turns a shape around a centre by a given angle and direction.
- ●A dilation multiplies all lengths by a scale factor, producing a similar (not congruent) shape.
- ●Translations, reflections, and rotations preserve size → congruent. Dilations change size → similar.