G7: Transformations
Identify, describe and construct congruent and similar shapes; rotation, reflection, translation, enlargement; fractional and negative scale factors
📋 Key Concepts
Definition: A transformation changes the position or size of a shape. The four types are: translation, reflection, rotation, and enlargement.
| Transformation | Effect | Size Changed? |
|---|---|---|
| Translation | Moves position | No |
| Reflection | Flips over a line | No |
| Rotation | Turns around a point | No |
| Enlargement | Changes size | Yes |
📝 Translation
Definition: A translation moves a shape without rotating or reflecting it. Described using a column vector.
Translation:
means move a units right, b units up
| a |
| b |
Example 1
Translate triangle ABC by vector
| 3 |
| -2 |
Solution:
Move each point 3 units right and 2 units down.
Point A (1, 4) → A' (1 + 3, 4 - 2) = (4, 2)
Point B (3, 4) → B' (6, 2)
Point C (2, 6) → C' (5, 4)
📝 Reflection
Definition: A reflection flips a shape over a line (mirror line). The reflected shape is congruent to the original.
Common mirror lines:
- x-axis: y = 0
- y-axis: x = 0
- y = x (diagonal)
- y = -x
- Vertical lines: x = a
- Horizontal lines: y = b
Example 2
Reflect point P (3, 5) in the line x = 2.
Solution:
Distance from point to line = 3 - 2 = 1 unit
Reflected point is 1 unit on the other side: 2 - 1 = 1
P' = (1, 5)
📝 Rotation
Key details needed:
- Centre of rotation
- Angle of rotation (usually 90°, 180°, or 270°)
- Direction (clockwise or anticlockwise)
Example 3
Rotate point A (4, 2) 90° clockwise about the origin.
Solution:
90° clockwise about origin: (x, y) → (y, -x)
A (4, 2) → A' (2, -4)
Tip: 90° anticlockwise: (x, y) → (-y, x)
180°: (x, y) → (-x, -y)
📝 Enlargement
Definition: An enlargement changes the size of a shape. The scale factor determines how much bigger or smaller.
If scale factor k:
- Length multiplied by k
- Area multiplied by k²
- Volume multiplied by k³
Example 4
Enlarge triangle ABC with scale factor 2, centre at (0, 0). A is at (1, 2).
Solution:
From centre (0, 0) to A (1, 2): distance is (1, 2)
Multiply by scale factor: (1 × 2, 2 × 2) = (2, 4)
A' is at (2, 4)
📝 Fractional Scale Factors
Fractional scale factors (0 < k < 1): The image is smaller than the object.
Example 5
Enlarge with scale factor 1/2, centre (0, 0). Point P is at (6, 4).
Solution:
From centre to P: (6, 4)
Multiply by 1/2: (6 × 1/2, 4 × 1/2) = (3, 2)
P' is at (3, 2)
📝 Negative Scale Factors
Negative scale factors: The image is on the opposite side of the centre, AND the size changes.
Example 6
Enlarge with scale factor -2, centre (1, 1). Point A is at (2, 3).
Solution:
From centre to A: (2 - 1, 3 - 1) = (1, 2)
Multiply by -2: (1 × -2, 2 × -2) = (-2, -4)
Add centre: (-2 + 1, -4 + 1) = (-1, -3)
A' is at (-1, -3)
❓ Practice Questions
Q1: Translate point (3, 7) by vector
| -2 |
| 4 |
Q2: Reflect point (5, 3) in the y-axis.
Q3: Rotate point (2, 5) 180° about the origin.
Q4: Enlarge point (4, 6) with scale factor 3, centre (0, 0).
Q5: Enlarge point (8, 4) with scale factor -1/2, centre (0, 0).
✅ Answers
- (1, 11)
- (-5, 3)
- (-2, -5)
- (12, 18)
- (-4, -2)
🎯 Exam Tips
- Always state the transformation type first when describing
- For translations, use column vector notation
- For rotations, give centre, angle, and direction
- For enlargements, give centre and scale factor
- Use tracing paper in the exam to help visualise
- Count squares carefully when finding coordinates
🧠 Problem-Solving Strategies
Problem-Solving
For reflections: identify the mirror line equation. For rotations: state centre, angle and direction. For translations: give the column vector. For enlargements: state centre and scale factor. Always describe transformations FULLY and check by seeing if corresponding points map correctly.
Multi-Step Problem
Triangle A has vertices (1,2), (3,2), (3,5). It is enlarged by scale factor 2 with centre (0,0). Find the coordinates of the image, then find the area of both triangles and compare.
Solution: Image vertices: (2,4), (6,4), (6,10). Original area = 1/2 x 2 x 3 = 3 square units. Image area = 1/2 x 4 x 6 = 12 square units. Area ratio = 12/3 = 4 = 2 squared. The area scale factor is the square of the linear scale factor.
⚠️ Common Errors
Watch Out!
1. Wrong: Describing a rotation as 90 degrees without stating clockwise or anticlockwise Correct: State direction explicitly: "90 degrees clockwise" or "90 degrees anticlockwise". Both give different results.
2. Wrong: For enlargement with fractional scale factor, moving vertices towards the centre instead of using the correct ratio Correct: For SF 1/2 with centre (2,3), a point at (6,7): distance from centre = (4,4), halve to (2,2), image at (2+2, 3+2) = (4,5).
3. Wrong: Describing a translation as "2 right, 3 up" instead of using a column vector Correct: Write as a column vector with 2 on top and 3 below. This is the required mathematical notation.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Shape S has vertices at (1,1), (3,1), (3,3), (1,3). (a) Reflect S in the line y = x. Call the image S-prime. State the coordinates of S-prime. (b) Rotate S-prime 90 degrees anticlockwise about (0,0). Call the image S-double-prime. State the coordinates of S-double-prime. (c) Describe the single transformation that maps S directly to S-double-prime.
(a) Reflection in y = x swaps x and y coordinates. S-prime has vertices: (1,1), (1,3), (3,3), (3,1).
(b) Rotation 90 degrees anticlockwise about origin maps (x,y) to (-y, x). S-double-prime has vertices: (-1,1), (-3,1), (-3,3), (-1,3).
(c) The single transformation from S to S-double-prime is a reflection in the y-axis. (x,y) maps to (-x,y). Check: (1,1) becomes (-1,1), (3,1) becomes (-3,1), (3,3) becomes (-3,3), (1,3) becomes (-1,3). Verified.
Mark scheme: M1 reflection method, A1 correct coordinates, M1 rotation method, A1 correct coordinates, M1 identifying single transformation, A1 reflection in y-axis
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A shape is enlarged by scale factor 3 with centre of enlargement at (1,1). The original shape has area 5 cm squared.
(a) What is the area of the image?
(b) A point at (4,5) is on the original shape. Find the coordinates of its image.
(c) A student says "Enlargement always makes a shape bigger." Is this correct? Give an example to support your answer.
Answers: (a) Area scale factor = 3 squared = 9. Image area = 5 x 9 = 45 cm squared. (b) Distance from centre = (4-1, 5-1) = (3,4). Multiply by 3: (9,12). Image = (1+9, 1+12) = (10,13). (c) No — if the scale factor is between 0 and 1, the image is SMALLER than the original. For example, SF = 0.5 produces an image half the size.
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🎬 Video Resources
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