Transformations

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Notes

Translations

A **translation** moves a shape without changing its size or orientation; the object and image are **congruent**.
Movement is described by a **column vector** $\begin{pmatrix} x \\ y \end{pmatrix}$ where $x$ is horizontal (right positive, left negative) and $y$ is vertical (up positive, down negative).
To translate a shape, move each vertex by the vector and join the new vertices.
To describe a translation, state 'translation' and give the vector.
To reverse a translation, use the same vector with both signs changed.

Reflections

A **reflection** flips a shape across a **mirror line** (line of reflection); the image is congruent and the same distance from the line as the object.
Points on the mirror line are **invariant** (do not move).
To reflect a shape, measure perpendicular distance from each vertex to the mirror line and plot the same distance on the opposite side.
To describe a reflection, state 'reflection' and give the equation of the mirror line (e.g., $x = k$ , $y = k$ , $y = x$ , $y = -x$ ).
To reverse a reflection, apply the same reflection again.

Rotations

A **rotation** turns a shape about a fixed **centre of rotation**; the image is congruent.
You need the centre, angle (90°, 180°, 270°), and direction (clockwise or anti-clockwise). For 180°, direction is not needed.
To rotate a shape, use tracing paper: trace the shape, place pencil on centre, rotate by the angle, and draw the image.
To describe a rotation, state 'rotation', centre, angle, and direction.
To reverse a rotation, rotate by the same angle in the opposite direction about the same centre.

Enlargements

An **enlargement** changes the size of a shape by a **scale factor** (SF) from a **centre of enlargement** (CoE).
If $SF > 1$ , the image is larger; if $0 < SF < 1$ , the image is smaller (fractional enlargement).
To enlarge a shape, multiply horizontal and vertical distances from CoE to each vertex by SF, then plot the new vertices.
To describe an enlargement, state 'enlargement', SF, and CoE coordinates.
To reverse an enlargement, use the reciprocal SF with the same CoE.

Combined Transformations & Describing Fully

A **single transformation** maps one shape to another; you must state the type and all required details (vector, mirror line, centre/angle/direction, or SF and CoE).
When describing, always include: type of transformation, and the specific parameters (e.g., 'translation by vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ ').
Use tracing paper to check rotations and reflections; draw lines from CoE through vertices to verify enlargements.

Translation Example

Reflection Example

Rotation Example

Enlargement Example

Practice questions

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1.What is the name of the transformation that flips a shape across a line?
Easy
- AReflection
- BRotation
- CTranslation
- DEnlargement
2.Under a translation, what remains the same about the shape?
Easy
- ASize and orientation
- BSize only
- COrientation only
- DPosition
3.What is the scale factor if a shape is enlarged to twice its original size?
Easy
- A2
- B $\frac{1}{2}$
- C1
- D4
4.A translation is described by a vector of the form $\begin{pmatrix} x \\ y \end{pmatrix}$ . What does a negative value of $x$ represent?
Easy
- AMove left
- BMove right
- CMove down
- DMove up
5.When rotating a shape, what is the point about which the shape turns called?
Easy
- ACentre of rotation
- BCentre of enlargement
- CMirror line
- DTranslation vector
6.A shape is translated by vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ . Which of the following describes the reverse translation?
Medium
- A $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$
- B $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
- C $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$
- D $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$
7.A shape is reflected in the line $y = x$ . What is the image of the point (2, 5)?
Medium
- A(5, 2)
- B(2, -5)
- C(-2, 5)
- D(-5, -2)
8.A triangle with vertices at (1, 2), (3, 2), (2, 5) is rotated 90° clockwise about the origin. What are the coordinates of the image of (1, 2)?
Medium
- A(2, -1)
- B(-2, 1)
- C(-1, -2)
- D(1, -2)

Summary

## Transformations (IGCSE Core) ### 1. Translations - **Definition:** Moves a shape without changing size or orientation. Object and image are **congruent**. - **Vector:** Column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ where $x$ = horizontal movement (right positive, left negative), $y$ = vertical movement (up positive, down negative). - **To describe:** State "translation" and give the vector. - **Reverse:** Change signs of both components. ### 2. Reflections - **Definition:** Flips a shape across a **mirror line** (line of reflection). Object and image are congruent. - **Mirror lines:** Vertical $x = k$, horizontal $y = k$, diagonal $y = x$ or $y = -x$. - **To describe:** State "reflection" and the equation of the mirror line. - **Reverse:** Same reflection again. ### 3. Rotations - **Definition:** Turns a shape about a **centre of rotation** by a given angle and direction. Object and image are congruent. - **Angle:** Usually 90°, 180°, or 270°. Direction: clockwise or anti-clockwise (omit for 180°). - **To describe:** State "rotation", centre, angle, and direction. - **Reverse:** Same centre, same angle, opposite direction. ### 4. Enlargements - **Definition:** Changes size by a **scale factor** (SF) about a **centre of enlargement** (CoE). Object and image are **similar**. - **Scale factor:** $\text{SF} = \frac{\text{image length}}{\text{object length}}$. SF > 1 enlarges; 0 < SF < 1 reduces. - **To describe:** State "enlargement", SF, and CoE coordinates. - **Reverse:** Enlargement with reciprocal SF and same CoE. **Tip:** Use tracing paper for rotations and reflections. Always check corresponding points align through CoE for enlargements.

Key terms

Translation: A transformation that moves a shape by a given vector without changing its size or orientation.
Reflection: A transformation that flips a shape across a line (mirror line) to produce a mirror image.
Rotation: A transformation that turns a shape about a fixed point (centre of rotation) by a given angle and direction.
Enlargement: A transformation that changes the size of a shape by a scale factor about a centre of enlargement.
Scale factor: The ratio of a length on the image to the corresponding length on the object; determines how much an enlargement changes size.
Centre of enlargement: The fixed point from which distances are measured in an enlargement; lines through corresponding vertices meet here.
Mirror line: The line over which a shape is reflected; also called line of reflection.
Centre of rotation: The fixed point about which a shape is rotated.
Column vector: A vector written as $\begin{pmatrix} x \\ y \end{pmatrix}$ describing horizontal (x) and vertical (y) movement in a translation.
Congruent: Shapes that are identical in size and shape; translations, reflections, and rotations produce congruent images.
Similar: Shapes that have the same angles and proportional sides; enlargements produce similar images.
Invariant point: A point that does not move under a transformation; e.g., points on the mirror line in a reflection.

Slides

What are Transformations?

Transformations change the position, orientation, or size of a shape.
The original shape is called the object; the new shape is the image.
There are four types: translations, reflections, rotations, and enlargements.
Corresponding vertices are labelled A, B, C... on the object and A', B', C'... on the image.

What is the name for the original shape before a transformation?

Image
Object
Vertex
Centre

Translations

A translation moves a shape without changing its size or orientation.
The object and image are congruent.
Movement is described by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ .
x: horizontal movement (right positive, left negative). y: vertical movement (up positive, down negative).

A translation vector $\begin{pmatrix} -3 \\ 4 \end{pmatrix}$ means move ____ units left $and ____$ units up.

Reflections

A reflection flips a shape across a mirror line (line of reflection).
The image is the same size, but reversed orientation.
Each point and its image are the same perpendicular distance from the mirror line.
Points on the mirror line are invariant (do not move).

What is the equation of the mirror line for a reflection that maps shape A to shape B?

$x = 1$
$y = 1$
$y = x$
$y = -x$

Rotations

A rotation turns a shape about a fixed point called the centre of rotation.
The image is the same size and shape, but in a new orientation.
You need to know: centre, angle (90°, 180°, 270°), and direction (clockwise or anticlockwise).
For 180°, direction is not needed.

Which of these are true about rotations?

The centre of rotation is always inside the shape.
A rotation of 90° clockwise is the same as 270° anticlockwise.
Tracing paper can help find the centre of rotation.

Enlargements

An enlargement changes the size of a shape by a scale factor (SF).
If $SF > 1$ , the image is larger. If $0 < SF < 1$ , the image is smaller (fractional enlargement).
The centre of enlargement (CoE) is a fixed point from which distances are scaled.
Lines from the CoE through vertices of the object pass through corresponding vertices of the image.

A shape is enlarged with scale factor 3 and centre (0,0). The image is:

Three times as far from the centre
One third as far from the centre
The same distance
Moved without scaling

Describing Transformations

To describe a translation: state 'translation' and give the column vector.
To describe a reflection: state 'reflection' and give the equation of the mirror line.
To describe a rotation: state 'rotation', centre, angle, and direction.
To describe an enlargement: state 'enlargement', scale factor, and centre of enlargement.

To reverse a translation with vector $\begin{pmatrix} 2 \\ -5 \end{pmatrix}$ , use the vector $\begin{pmatrix} ____ \\ ____ \end{pmatrix}$ .

Reversing Transformations

To reverse a reflection, reflect again in the same mirror line.
To reverse a rotation, rotate by the same angle in the opposite direction about the same centre.
To reverse an enlargement, use the reciprocal scale factor with the same centre.
For a translation, reverse the signs of the vector components.

A shape is enlarged by scale factor 1/2. What scale factor reverses this?

$\frac{1}{2}$
2
-2
1

Fractional Enlargements

A fractional scale factor (between 0 and 1) makes the image smaller.
The image is closer to the centre of enlargement than the object.
Multiply distances from the centre by the fraction.
Example: $SF = \frac{1}{2}$ means distances are halved.

Which of these are true about fractional enlargements?

The image is always smaller than the object.
The centre of enlargement is always inside the object.
The scale factor can be written as a fraction.

Summary: Key Points

Translations: vector movement, congruent image.
Reflections: mirror line, perpendicular distances equal.
Rotations: centre, angle, direction (clockwise/anticlockwise).
Enlargements: scale factor, centre of enlargement, size change.

Which transformation always produces a congruent image?

Enlargement
Translation
Both translation and reflection
All transformations

Flashcards

What are the four types of transformations in maths?

Translations, rotations, reflections, and enlargements.

What is a translation?

A translation moves a shape without changing its size or orientation. The object and image are congruent.

How is a translation vector written?

As a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ , where $x$ is the horizontal movement (positive right, negative left) and $y$ is the vertical movement (positive up, negative down).

Translate shape P by vector $\begin{pmatrix} -4 \\ 5 \end{pmatrix}$ . What does this mean?

Move the shape 4 units to the left and 5 units up.

How do you describe a translation?

State that the transformation is a translation and give the column vector that describes the movement.

How do you reverse a translation?

Reverse the signs of both values in the column vector. For example, the reverse of $\begin{pmatrix} -2 \\ 7 \end{pmatrix}$ is $\begin{pmatrix} 2 \\ -7 \end{pmatrix}$ .

What is a reflection?

A reflection flips a shape across a mirror line (line of reflection). The image is the same size as the object but reversed.

How do you reflect a shape in a vertical or horizontal line?

Measure the perpendicular distance from each vertex to the mirror line, then plot the reflected point the same distance on the opposite side. Connect the points.

How do you describe a reflection?

State that the transformation is a reflection and give the equation of the mirror line (e.g., $x = k$ , $y = k$ , $y = x$ , or $y = -x$ ).

How do you reverse a reflection?

Reflect the shape again in the same mirror line. The reverse transformation is the same reflection.

What is a rotation?

A rotation turns a shape around a fixed point called the centre of rotation. The image is the same size as the object.

How do you rotate a shape by 90° clockwise about a point?

Place tracing paper over the shape, mark the centre of rotation, rotate the paper 90° clockwise, and draw the image in the new position.

How do you describe a rotation?

State that it is a rotation, give the centre of rotation, the angle (90°, 180°, or 270°), and the direction (clockwise or anti-clockwise). For 180°, direction is not needed.

How do you reverse a rotation?

Rotate the shape by the same angle in the opposite direction about the same centre.

What is an enlargement?

An enlargement changes the size of a shape by a scale factor, relative to a centre of enlargement. The image is similar to the object.

How do you enlarge a shape with scale factor 2 and centre (2,1)?

From the centre, count the horizontal and vertical distances to each vertex, multiply by 2, and plot the new vertices. Connect them to form the enlarged shape.

How do you describe an enlargement?

State that it is an enlargement, give the scale factor, and the coordinates of the centre of enlargement.

How do you reverse an enlargement?

Enlarge the image by the reciprocal of the scale factor using the same centre of enlargement.

What is a fractional enlargement?

An enlargement with a scale factor between 0 and 1. The image is smaller than the object and closer to the centre of enlargement.

How do you find the scale factor of an enlargement?

Divide the length of a side on the image by the length of the corresponding side on the object.

What are invariant points in transformations?

Points that do not move during a transformation. For example, points on the mirror line in a reflection, or the centre of rotation in a rotation.

Describe fully the transformation that maps shape A onto shape B if B is a translation of A by vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ .

Translation by vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ .

Describe fully the transformation that maps shape A onto shape B if B is a reflection of A in the line $x = -1$ .

Reflection in the line $x = -1$ .

Describe fully the transformation that maps shape A onto shape B if B is a rotation of A by 90° clockwise about the origin.

Rotation of 90° clockwise about the origin.

Describe fully the transformation that maps shape A onto shape B if B is an enlargement of A by scale factor 3 with centre (9,7).

Enlargement by scale factor 3 with centre (9,7).

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