An enlargement is a transformation where an object maps to an image of the same shape but different size. The object and the image are said to be similar. An enlargement requires a centre of enlargement and a scale factor.

### Notation

E is an enlargement, with centre of enlargement O and scale factor µ.

E: AB A´B´

### Properties of Enlargement

Triangle PQR maps to triangle P´Q´R´ under enlargement with centre O.

Lines and their images are always parallel.

e.g. PQ is parallel to P´Q´

Angle size is invariant.

e.g. PQR = P´Q´R´

The centre of enlargement is the only invariant point.

Length and area are not invariant, except when m = 1 or − 1.

Enlargement is a direct transformation.

i.e. PQR and P´Q´R´ are both anti-clockwise.

If μ is the scale factor for length, mμ

^{ 2 }is the scale factor for area.

### Different Scale Factors

The diagram below shows the effect of a variety of scale factors on the enlargement of a triangle ABC about centre O.

- If the scale factor is positive, both the object and the image are on the same side of the centre.
- If the scale factor is negative, the object and the image are on opposite sides of the centre.

The image is inverted.

See examples of enlargements −

### Location of the Centre of Enlargement

Given a figure and its image, to find the centre of enlargement:

1. Join up a point and its image.

2. Repeat for another point and its image.

3. The centre of enlargement is the intersection of these lines.

**Similar Figures**

Figures are similar if they have the same shape.

- Similar figures can be mapped onto one another by an enlargement or by a combination of reflection, rotation or translation and an enlargement.
- The corresponding angles of similar figures are equal.
- The corresponding sides are proportional to one another.

**Type 1**

YXZ is similar to PQM as they have corresponding angles equal.

P = Y |

YXZ can be mapped to PQM by a combination of transformations.

Scale factor = µ =

**Type 2**

ABC is similar to ADE because:

A is common, (DE is parallel to BC) |

ABC can be mapped onto ADE by an enlargement.

Examples |
Answer |

(a) (i) The two triangles A and B are similar. Find p and q. |
(a) (i) For A B , scale factor = 2 p = 2 × 10 p = 20 2 x q = 12 q = 6 |

(ii) The area of triangle A is 24 units |
(ii) Scale factor for area = µ Area of triangle B = 4 x area of triangle A |

(b) Find x, if BD is parallel to CE |
(b) For ABD ACE 12x = 8(x + 6) 12x = 8x + 48 4x = 48 x = 12 |

Download an interactive spreadsheet (Microsoft Excel) showing reflections, rotations, translations and **enlargements**.

(Windows users, right click and "Save target as..." to save the files on your computer.)