Maths

Transformations

Transforming an object means changing its appearance. There are many ways to do this, but we are going to concentrate on just four of them.

In each of these sections, we refer to the original shape as the **object** and the transformed shape as the **image**.

If we **translate**
an object, we move it up or down
or from side to side. But we **do not**
change its shape, size or direction.

- Question
Which of the following triangles (P, Q or R) is a translation of triangle ABC?

- Answer
Q

**is**a translation of triangle ABC. Triangle ABC has been translated 3 squares to the right and 2 squares up.P

**is not**a translation of triangle ABC because the object and its image are facing in opposite directions.R

**is not**a translation of triangle ABC because the object and its image have different side lengths. They are different shapes.

**Remember**

When we translate an object, every vertex (corner) must be moved in the same way.

**Example:**

Triangle PQR has been translated 3 squares down and 4 squares to the right. All of the vertices have been translated in the same way, and the object and its image are exactly the same shape and size.

The line **AB** has been translated 4 units to the right and 2 units upwards. We say that the displacement vector is .

We always write the **horizontal displacement** at the **top**
of the vector and the **vertical displacement** at the **bottom**.

A move downwards or to the left is indicated by a **-** sign:

The object X has been translated by the vector
to give its image, **X**'.

- Question
**P'**(7, 2) is the image of**P**(2, 11) after a translation. What vector describes this translation?

- Answer
Did you get ?

If so - well done.

To work out the horizontal movement after the translation, subtract the x-coordinates for p from p':

7 - 2 = 5

To work out the vertical movement after the translation, subtract the y-coordinates for p from p':

2 - 11 = - 9

The vector that describes the translation is:

If you are not sure, try drawing a sketch and marking the points

**P**and**P'**on it.The shape has been translated 5 units to the right and 9 units downwards.

Remember that any move to the left or downwards needs a

**-**sign.

When an object is transformed by a reflection the object and its image are always the same perpendicular distance from the mirror line.

Perpendicular means 'at right angles to'.

**Examples:**

A and A' are the same perpendicular distance from the mirror line, as are B and B' and C and C'.

The object (ABC) has been **reflected** in the mirror line to give the image (A'B'C').

P and P' are the same perpendicular distance from the mirror line, as are Q and Q', R and R' and S and S'.

The object (PQRS) has been **reflected** in the mirror line to give the image (P'Q'R'S').

In your exam, you might have to reflect a shape in a given line. You will need to remember that:

**y = c** describes a **horizontal** line through c.

**x = c** describes a **vertical** line through c.

A point and its image are always the same perpendicular distance from the mirror line. Have a look at the reflections below:

Look at this table:

Coordinates of vertices | Reflection in y = x | Reflection in y = -x | |
---|---|---|---|

A | (1, 2) | (2, 1) | (-2, -1) |

B | (1, 5) | (5, 1) | (- 5, -1) |

C | (3, 2) | (2, 3) | (-2, -3) |

Can you see the pattern?

After a reflection in the line **y = x**, the x and y coordinates
**swap over**.

After a reflection in the line **y = - x**, the x and y coordinates
**swap over and change signs**.

We can rotate an object ^{1}/_{4} turn, ^{1}/_{2} turn or ^{3}/_{4} turn around a given centre.

**Example**

You may also be asked to rotate an object **around the origin**. Practise **rotations** using the transformations activity.

Remember that a point and its image are always the same distance from the centre of rotation.

**Example**

Describe the rotation which moves **ABCD** to **A'B'C'D'** in this diagram.

The **centre of rotation**
is **(0, -1)**.

We can see that **A** is the same distance from (0, -1)
as **A'**.

**D** is the same distance from (0, -1) as **D'**.

The **angle of rotation** is **90° clockwise**.

It is possible to use constructions to find the centre of rotation,
but this can be time-consuming and complicated. Try to find the
centre by observation, but remember that each point and its image
**must** be the same distance from the centre of rotation.

When working out enlargements, you will need to know the **scale factor** and **centre of enlargement**.

- The
**scale factor**tells us by how much the object has been enlarged. - The
**centre of enlargement**tells us where the enlargement is being measured from.

**Example:**

P'Q' = 3 × PQ

Q'R' = 3 × QR

P'R' = 3 × PR

Therefore, we say that the transformation is an **enlargement** with **scale factor 3**.

You may be asked to enlarge a shape with a given **centre of enlargement**. Look through this slideshow to see an example.

**To enlarge an object, scale factor 2.**

The centre of enlargement is O.

Draw lines from the centre of enlargement.

OA' is twice as far as OA, because the scale factor is 2.

Draw the image shape.

**To enlarge an object, scale factor 3.**

This method is identical to the previous one, but

OA' = 3 × OA

OB' = 3 × OB

OC' = 3 × OC

The centre of enlargement is not always outside the object. Look at the following examples:

The next diagram shows an enlargement, scale factor 3. The centre of enlargement, X, is also a point of the object WXYZ. The result is that X' is in the same place as X.

The next diagram shows an enlargement, scale factor 2, centre O, where O is inside the object. The result is that the image surrounds the object.