Enlargement

Babushka_dolls.JPGAn 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
Properties
Scale Factors
Location of Centre
Similar Figures
Examples

Notation

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

E: AB Y10_Enlargement_01.gifA´B´Y10_Enlargement_02.gif

Y10_Enlargement_03.gif

Y10_Enlargement_04.gif

 

Properties of Enlargement

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

Y10_Enlargement_05.gif

Lines and their images are always parallel.

e.g. PQ is parallel to P´Q´

Angle size is invariant.

e.g. Y10_Enlargement_06.gifPQR = Y10_Enlargement_06.gifP´Q´R´

The centre of enlargement is the only invariant point.

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

i.e. Enlargement is not an isometry.

Enlargement is a direct transformation.

i.e. Y10_Enlargement_07.gifPQR and Y10_Enlargement_07.gifP´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.

Y10_Enlargement_08.gif
  • 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 − button_animation.gif

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

Y10_Enlargement_07.gifYXZ is similar to Y10_Enlargement_07.gifPQM as they have corresponding angles equal.

 

Y10_Enlargement_06.gifP = Y10_Enlargement_06.gifY
Y10_Enlargement_06.gifQ = Y10_Enlargement_06.gifX
Y10_Enlargement_06.gifM = Y10_Enlargement_06.gifZ
Y10_Enlargement_09.gif

 

Y10_Enlargement_07.gifYXZ can be mapped to Y10_Enlargement_07.gifPQM by a combination of transformations.

Scale factor = µ = Y10_Enlargement_10.gif

Type 2

Y10_Enlargement_07.gifABC is similar to Y10_Enlargement_07.gifADE because:

 

Y10_Enlargement_06.gifA is common,
Y10_Enlargement_06.gifB = Y10_Enlargement_06.gif D
Y10_Enlargement_06.gifC = Y10_Enlargement_06.gifE

(DE is parallel to BC)
Y10_Enlargement_11.gif

Y10_Enlargement_07.gifABC can be mapped onto Y10_Enlargement_07.gifADE by an enlargement.

Scale factor = µ = Y10_Enlargement_12.gif

 

Examples
Answer

(a) (i) The two triangles A and B are similar. Find p and q.Y10_Enlargement_13.gif

(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. 
What is the area of triangle B?

(ii) Scale factor for area = µ2 = 4

Area of triangle B = 4 x area of triangle A
= 4 x 24 = 96 units2

(b) Find x, if BD is parallel to CE Y10_Enlargement_14.gif

(b) For ABD ACE

Y10_Enlargement_15.gif
Y10_Enlargement_16.gif
12x = 8(x + 6)
12x = 8x + 48
4x = 48
x = 12

 

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

button_download.gif

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