Enlargements, Ratio and Scale Factors

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.

 

 

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

 

Question 2