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

Worked examples

Question 2

The shape ABCD has been enlarged to A'B'C'D'. What is the scale factor?

Think: 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

Do: By aligning the largest lengths sides with each other AD and A'D', and then the other sides we 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.

Question 3

Is shape $ABCD$ABCD a reduction of shape $SPQR$SPQR?

Think: Firstly, we need to identify corresponding sides. To do this we will rotate $SPQR$SPQR.

Do: Now we 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 $ABCD$ABCD is not a reduction of $PQRS$PQRS.

Question 4

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$324 m tall, and on this image the tower is $23.5$23.5 cm, what is the scale factor?

Do:

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

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$1 cm$:$: $1378.72$1378.72 cm

Practice problems

Question 5

Which of these shapes are enlargements of each other?

A

B

C

D

Question 6

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

Two similar triangles are depicted, with the larger one labeled ABC and the smaller one labeled A'B'C'. They are positioned so that the extensions of their corresponding sides, BB' and CC', converge at a shared point, O. The line segment OB is $5$5 units in length, and BB' extends an additional $15$15 units, resulting in a total length of $20$20 units for OB'. In a similar fashion, OC is $5$5 units long, and the extension CC' adds $15$15 units, totaling $20$20 units for OC'. Point O is the vertex of two isosceles triangles, BOC and B'OC', with the bases of these triangles being defined by the vertices of the larger and smaller triangles, respectively.

$\frac{1}{4}$14
A
$3$3
B
$\frac{1}{3}$13
C
$4$4
D

Outcomes

II.G.SRT.2

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

8.03 Similar figures

Enlargements

Worked example

Question 1

Reductions