CIE IGCSE Mathematics (0580) Transformations Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Transformations Study Notes
LEARNING OBJECTIVE
- Understanding the idea of different types Transformation of Shapes
Key Concepts:
- Reflection of Shapes
- Rotation of Shapes
- Enlargement of Shapes
- Translation of Shapes
Transformation: Reflection of Shapes
Transformation: Reflection
Reflection is a transformation that flips a shape over a specific line, called the mirror line. The shape appears as a mirror image on the opposite side of the line. Reflections preserve size and shape (the reflected shape is congruent to the original).
Reflection in Vertical or Horizontal Lines:
- A vertical mirror line has the equation \( x = k \) where \( k \) is a constant. Every point’s reflected x-coordinate changes according to its distance from this line, while the y-coordinate stays the same.
- A horizontal mirror line has the equation \( y = k \) where \( k \) is a constant. Every point’s reflected y-coordinate changes according to its distance from this line, while the x-coordinate stays the same.
Key Properties:
- Reflected shapes are congruent to the original shape.
- Each point and its image are the same perpendicular distance from the mirror line.
- Orientation is reversed in the direction perpendicular to the mirror line.
Example:
A triangle is drawn with vertices at A(2, 1), B(4, 1), and C(3, 3). Reflect the triangle in the vertical line \( x = 5 \). Give the coordinates of the reflected image.
▶️ Answer/Explanation
To reflect across the line \( x = 5 \), calculate how far each x-coordinate is from 5, and reflect it across.
A(2, 1) → A′(8, 1)
B(4, 1) → B′(6, 1)
C(3, 3) → C′(7, 3)
Answer: Reflected triangle has vertices at A′(8, 1), B′(6, 1), C′(7, 3).
Example:
A square has vertices at P(1, 3), Q(1, 6), R(4, 6), and S(4, 3). Reflect the square in the horizontal line \( y = 4 \). Give the coordinates of the image.
▶️ Answer/Explanation
Reflect across \( y = 4 \): measure vertical distance of each vertex from line \( y = 4 \), then reflect it symmetrically.
P(1, 3) → P′(1, 5)
Q(1, 6) → Q′(1, 2)
R(4, 6) → R′(4, 2)
S(4, 3) → S′(4, 5)
Answer: Reflected square has vertices at P′(1, 5), Q′(1, 2), R′(4, 2), S′(4, 5).
Transformation: Rotation of Shapes
Transformation: Rotation
Rotation is a transformation where a shape is turned around a fixed point (called the center of rotation) through a specific angle and direction (clockwise or anticlockwise). The size and shape of the figure do not change — only the position changes.
Key Points:
- Common rotation angles: 90°, 180°, 270°, 360°.
- Clockwise and anticlockwise directions must be specified unless otherwise stated.
- A shape can be rotated about any point: origin (0, 0), a vertex, or the midpoint of an edge.
- Rotated shapes are congruent to the original shape.
Rotation about the Origin — Coordinate Rules:
- 90° anticlockwise: \( (x, y) \rightarrow (-y, x) \)
- 180°: \( (x, y) \rightarrow (-x, -y) \)
- 90° clockwise: \( (x, y) \rightarrow (y, -x) \)
Example:
A rectangle has vertices at A(1, 2), B(4, 2), C(4, 5), and D(1, 5). Rotate the rectangle 90° clockwise about the origin (0, 0). Find the coordinates of the image.
▶️ Answer/Explanation
Use rotation rule: 90° clockwise about the origin: \( (x, y) \rightarrow (y, -x) \)
A(1, 2) → A′(2, -1)
B(4, 2) → B′(2, -4)
C(4, 5) → C′(5, -4)
D(1, 5) → D′(5, -1)
Answer: Image rectangle has vertices A′(2, -1), B′(2, -4), C′(5, -4), D′(5, -1).
Example:
A triangle has vertices at P(3, 1), Q(5, 1), and R(4, 4). Rotate the triangle 180° about the point (0, 0).
▶️ Answer/Explanation
Rotation rule: 180° about the origin: \( (x, y) \rightarrow (-x, -y) \)
P(3, 1) → P′(-3, -1)
Q(5, 1) → Q′(-5, -1)
R(4, 4) → R′(-4, -4)
Answer: Rotated triangle has vertices P′(-3, -1), Q′(-5, -1), R′(-4, -4).
Example:
Rotate a triangle with vertices A(3, 2), B(5, 2), and C(4, 4) 90° clockwise about the point (4, 2). Find the coordinates of the image.
▶️ Answer/Explanation
Step 1: Translate the shape so the center of rotation (4, 2) becomes the origin.
- A becomes \( (3 – 4, 2 – 2) = (-1, 0) \)
- B becomes \( (5 – 4, 2 – 2) = (1, 0) \)
- C becomes \( (4 – 4, 4 – 2) = (0, 2) \)
Step 2: Rotate each point 90° clockwise around the origin.
- \( (-1, 0) \rightarrow (0, 1) \)
- \( (1, 0) \rightarrow (0, -1) \)
- \( (0, 2) \rightarrow (2, 0) \)
Step 3: Translate the image points back by adding (4, 2):
- A′ = \( (0 + 4, 1 + 2) = (4, 3) \)
- B′ = \( (0 + 4, -1 + 2) = (4, 1) \)
- C′ = \( (2 + 4, 0 + 2) = (6, 2) \)
Answer: The rotated triangle has vertices A′(4, 3), B′(4, 1), C′(6, 2).
Transformation: Enlargement of Shapes
Transformation: Enlargement
Enlargement is a transformation that changes the size of a shape, while preserving its shape and angle measures. The shape is enlarged (or reduced) from a fixed point known as the centre of enlargement, using a specified scale factor.
Key Concepts:
- Centre of enlargement: the fixed point from which distances are measured.
- Scale factor \( k \):
- If \( k > 1 \), the image is an enlargement.
- If \( 0 < k < 1 \), the image is a reduction.
- If \( k < 0 \), the image is on the opposite side of the centre.
- All lengths in the image are multiplied by the scale factor.
- The angles and shape remain the same; only the size changes.
Coordinate Rule:
If a point \( (x, y) \) is enlarged by scale factor \( k \) from the origin (0, 0), then the image is \( (kx, ky) \).
Example:
A triangle has vertices at A(1, 1), B(3, 1), and C(2, 4). Enlarge the triangle by a scale factor of 2 about the origin (0, 0). Give the coordinates of the image.
▶️ Answer/Explanation
Multiply each coordinate by the scale factor (2):
A(1, 1) → A′(2, 2)
B(3, 1) → B′(6, 2)
C(2, 4) → C′(4, 8)
Answer: The enlarged triangle has vertices A′(2, 2), B′(6, 2), and C′(4, 8).
Example:
A square has vertices at P(2, 2), Q(4, 2), R(4, 4), and S(2, 4). Enlarge the square by a scale factor of \( \frac{1}{2} \) about the origin (0, 0).
▶️ Answer/Explanation
Multiply each coordinate by \( \frac{1}{2} \):
P(2, 2) → P′(1, 1)
Q(4, 2) → Q′(2, 1)
R(4, 4) → R′(2, 2)
S(2, 4) → S′(1, 2)
Answer: The reduced square has vertices P′(1, 1), Q′(2, 1), R′(2, 2), and S′(1, 2).
Example:
A triangle has vertices at A(2, 1), B(4, 1), and C(3, 3). Enlarge the triangle by scale factor 2 about the point (1, 1). Find the coordinates of the image.
▶️ Answer/Explanation
Use the enlargement formula:
If center is \( (x_0, y_0) \) and point is \( (x, y) \), then:
Image point = \( (x_0 + k(x – x_0), \ y_0 + k(y – y_0)) \)
Here, center = (1, 1), scale factor \( k = 2 \)
A(2, 1):
\( x’ = 1 + 2(2 – 1) = 1 + 2 = 3 \)
\( y’ = 1 + 2(1 – 1) = 1 + 0 = 1 \)
A′ = (3, 1)
B(4, 1):
\( x’ = 1 + 2(4 – 1) = 1 + 6 = 7 \)
\( y’ = 1 + 2(1 – 1) = 1 \)
B′ = (7, 1)
C(3, 3):
\( x’ = 1 + 2(3 – 1) = 1 + 4 = 5 \)
\( y’ = 1 + 2(3 – 1) = 1 + 4 = 5 \)
C′ = (5, 5)
Answer: The image has vertices A′(3, 1), B′(7, 1), and C′(5, 5).
Transformation: Translation of Shapes
Transformation: Translation
A translation moves every point of a shape the same distance in the same direction. The shape does not change in size, orientation, or shape.
Vector Notation:
A translation is represented using a column vector: \( \begin{pmatrix} x \\ y \end{pmatrix} \)
Where:
• \( x \): movement in the horizontal direction (right if positive, left if negative)
• \( y \): movement in the vertical direction (up if positive, down if negative)
Key Features:
- The shape remains congruent to the original.
- Orientation and size remain unchanged.
- Only position changes according to the vector.
Example:
Translate triangle A(1, 2), B(3, 2), C(2, 4) by the vector \( \begin{pmatrix} 4 \\ -1 \end{pmatrix} \).
▶️ Answer/Explanation
To translate a point by vector \( \begin{pmatrix} x \\ y \end{pmatrix} \), add \( x \) to the x-coordinate and \( y \) to the y-coordinate.
- A(1, 2) → \( (1+4, 2−1) = (5, 1) \)
- B(3, 2) → \( (3+4, 2−1) = (7, 1) \)
- C(2, 4) → \( (2+4, 4−1) = (6, 3) \)
Answer: A′(5, 1), B′(7, 1), C′(6, 3)
Example:
Translate a rectangle with corners at (2, 3), (6, 3), (6, 6), and (2, 6) by the vector \( \begin{pmatrix} -2 \\ 3 \end{pmatrix} \).
▶️ Answer/Explanation
- (2, 3) → \( (2 – 2, 3 + 3) = (0, 6) \)
- (6, 3) → \( (6 – 2, 3 + 3) = (4, 6) \)
- (6, 6) → \( (6 – 2, 6 + 3) = (4, 9) \)
- (2, 6) → \( (2 – 2, 6 + 3) = (0, 9) \)
Answer: The new rectangle has corners at (0, 6), (4, 6), (4, 9), (0, 9)