Enlargement or Dilatation

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Let us construct

Take any point x.

Join XA, XB, XC and produce them to points P, Q, R respectively, such that

XP = 3XA

XQ = 3XB

XR = 3XC

Join PQ, QR and PR

The sides of the are three times the corresponding sides of

The sides of the are parallel to the sides of

The point X is called the centre of enlargement.

Let us consider various cases for different values of the enlargement factor.

Enlargement factor greater than one.

when K > 1

Let K = 2

Take O as the centre of enlargement, two possible figures are shown:

(Pre-image)

AB = 3 cm

BC = 4.2 cm

AC = 6 cm

image

PQ = 6 cm

QR = 8.4 cm

PR = 12 cm

Enlargement factor less than one.

when K < 1

Let K = 1/3

In this case two possible figures are as shown:

(Pre-image)

AB = 4.5 cm

BC = 6 cm

AC = 7.5 cm

(Image)

PQ = 1.5 cm

QR = 2.0 cm

PR = 2.5 cm

Enlargement factor equal to one.

when K = 1

DABC (Pre-image) DPQR (Image)

AB = 2 cm PQ = 2 cm

BC = 3 cm QR = 3 cm

AC = 3.5 cm PR = 3.5 cm

To Summarise

(1) In case (iii) DABC is an image of itself.

(2) When K = 1, we get congruent figures and congruent figures are also similar.

(3) In each case, image is similar to pre-image.

(4) The ratio of corresponding sides is K.

(5) The corresponding sides are parallel to each other.

(6) The ratio of the areas of two similar figures is K².

(7) In solids, the ratio of the volumes of two similar solids will be K³.