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.
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.
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.
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.
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.
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 =
ABC is similar to ADE because:
A is common,
(DE is parallel to BC)
ABC can be mapped onto ADE by an enlargement.
(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 units2.
(ii) Scale factor for area = µ2 = 4
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.)