# Enlargements, Ratio and Scale Factors

Lesson

## Enlargements

A shape is considered an enlargement of another if one shape has side lengths that are all increased by the same scale factor.

#### For example:

Take a triangle with side lengths measuring $3$3cm, $4$4cm and $5$5cm. If each side is multiplied by the same factor, say $2$2, the new resulting triangle will have side lengths measuring $6$6cm, $8$8cm and $10$10cm. The resulting shape is larger.

## Reductions

A shape is considered a reduction of another if one shape has side lengths that are all decreased by the same scale factor.

Consider the reverse of the above example: a triangle with side lengths measuring $6$6cm, $8$8cm and $10$10cm has each side multiplied by a factor of $\frac{1}{2}$12. The new resulting triangle will have side lengths measuring $3$3cm, $4$4cm and $5$5cm. The resulting shape is smaller than the original.

Scale Factor!

The scale factor tells us by how much the object has been enlarged or reduced.

The scale factor can be greater than $1$1: image is being made bigger than the original.

The scale factor can be smaller than $1$1: image is being made smaller than the original.

##### Example 1

The shape ABCD has been enlarged to A'B'C'D'.

To find the scale factor we:

a) identify corresponding sides, in some cases this might mean rotating the shape.

b) look for a common multiple

By aligning the largest lengths sides with each other AD and A'D', and then the other sides I can set up this table.

Side Length Side Length Scale factor
AD $4$4 A'D' $12$12 $12\div4=3$12÷​4=3
DC $2$2 D'C' $6$6 $6\div2=3$6÷​2=3
CB $1$1 C'B' $3$3 $3\div1=3$3÷​1=3
BA $1$1 B'A' $3$3 $3\div1=3$3÷​1=3

Because shape A'B'C'D' has all side lengths $3$3 times larger than the corresponding sides of shape ABCD we say that it has been enlarged by a factor of $3$3

##### Example 2

Is shape ABCD an enlargement of shape SPQR?

Firstly, we need to identify corresponding sides.  To do this I will rotate SPQR.

Now I can see what might be the pairs of corresponding sides.

SIDE LENGTH SIDE LENGTH SCALE FACTOR
AB $2$2 PQ $10$10 $10\div2=5$10÷​2=5
BC $2$2 QR $10$10 $10\div2=5$10÷​2=5
CD $3$3 RS $12$12 $12\div3=4$12÷​3=4
DA $5$5 SP $15$15 $15\div5=3$15÷​5=3

As not all the sides have been decreased by the same scale, the shapes ABCD is not a reduction of PQRS.

##### example 3

Another place that enlargements are used is in scale drawings.  Consider this image of a plan of a tower.   If we know that the actual tower is $324$324m tall, and on this image the tower is $23.5$23.5cm we can actually deduce the scale factor.

 $\text{Scale factor }$Scale factor $=$= $\frac{\text{height of actual }}{\text{height of plan }}$height of actual height of plan ​ $=$= $\frac{324m}{23.5cm}$324m23.5cm​ $=$= $\frac{32400}{23.5}$3240023.5​ cm $=$= $1378.7234$1378.7234cm

So the actual tower is $1378.72$1378.72 times the height of the image on the paper.

We would write this as a scale of $1$1cm$:$: $1378.72$1378.72cm

#### Worked Examples

##### Question 1

Which of these shapes are enlargements of each other?

1. A

B

C

D

A

B

C

D

##### Question 2

Triangle A'B'C' has been reduced to form a smaller triangle ABC. What is the scale factor?

1. $\frac{1}{4}$14

A

$3$3

B

$\frac{1}{3}$13

C

$4$4

D

$\frac{1}{4}$14

A

$3$3

B

$\frac{1}{3}$13

C

$4$4

D