## Scale Factors Definition

In the above image, we have two triangles. Notice that the lengths of the triangle $A\text{'}B\text{'}C\text{'}$ are all exactly twice the lengths of the triangle $ABC$. Other than that, the triangles are exactly the same. Therefore, we can say that the two shapes are **similar** with a **scale** **factor** of **two**. We can also say that the side $AB$ **corresponds** to the side $A\text{'}B\text{'}$, the side $AC$ **corresponds** to the side $A\text{'}C\text{'}$ and the side $BC$ **corresponds** to the side $B\text{'}C\text{'}$.

A **scale factor** tells us the **factor** by which a shape has been **enlarged** by. The** corresponding sides** are the sides of the shape that have proportional lengths.

If we have a shape enlarged by a scale factor of three, then each side of the shape is multiplied by three to produce the new shape.

**Below is another example of a set of similar shapes. Can you work out the scale factor and corresponding sides? **

**Solution:**

We have two quadrilaterals $ABCD$ and $A\text{'}B\text{'}C\text{'}D\text{'}$. By looking at the shapes, we can see that $BC$ corresponds with $B\text{'}C\text{'}$ because they are both nearly identical- the only difference is $B\text{'}C\text{'}$ is longer. By how much?

Counting the squares, we can see that $BC$ is two units long, and $B\text{'}C\text{'}$ is six units long. To work out the scale factor, we divide the length of $BC$ by the length of $B\text{'}C\text{'}$. Thus, the scale factor is$\frac{6}{2}=3$.

**We can conclude that the scale factor is $3$ and the corresponding sides are $AB$ with $A\text{'}B\text{'}$, $BC$ with $B\text{'}C\text{'}$, $CD$ with $C\text{'}D\text{'}$ and $AD$ with $A\text{'}D\text{'}$. **

## Scale Factors Formulas

There is a very simple formula for working out the scale factor when we have two similar shapes. First, we need to identify the corresponding sides. Recall from earlier that these are the sides that are in proportion with each other. We then need to establish which is the **original** shape and which is the **transformed** shape. In other words, which is the shape that has been enlarged? This is usually stated in the question.

Then, we take an example of corresponding sides where the lengths of the sides are known and divide the length of the **enlarged** **side** by the length of the **original** **side**. This number is the **scale** **factor**.

Putting this mathematically, we have:

$SF=\frac{a}{b}$

Where $SF$ denotes the scale factor, $a$ denotes the enlarged figure side length and $b$ denotes the original figure side length and the side lengths taken are both from corresponding sides.

## Scale Factors Examples

In this section, we will look at some further scale factors examples.

**In the below image there are similar shapes $ABCDE$ and $A\text{'}B\text{'}C\text{'}D\text{'}E\text{'}$. We have: **

**$DC=16cm$, $D\text{'}C\text{'}=64cm$, $ED=xcm$, $E\text{'}D\text{'}=32cm$, $AB=4cm$ and $A\text{'}B\text{'}=ycm$. **

**Work out the value of $x$ and $y$.**

**Solution:**

Looking at the image, we can see that $DC$ and $D\text{'}C\text{'}$ are corresponding sides meaning that their lengths are in proportion with one another. Since we have the lengths of the two sides given, we can use this to work out the scale factor.

Calculating the scale factor, we have $SF=\frac{64}{16}=4$.

Thus, if we define $ABCDE$ to be the original shape, we can say that we can enlarge this shape with a scale factor of $4$ to produce the enlarged shape $A\text{'}B\text{'}C\text{'}D\text{'}E\text{'}$.

Now, to work out $x$, we need to work backwards. We know that $ED$ and $E\text{'}D\text{'}$ are corresponding sides. Thus, to get from $E\text{'}D\text{'}$ to $ED$ we must divide by the scale factor. We can say that $x=\frac{32}{4}=8cm$.

To work out y, we need to multiply the length of the side $AB$ by the scale factor. Thus, we have $A\text{'}B\text{'}=4\times 4=16cm$.

**Therefore $x=8cm$ and $y=16cm$. **

**Below are similar triangles $ABC$ and $A\text{'}B\text{'}C\text{'}$, both drawn to scale. Work out the scale factor to get from $ABC$ to $A\text{'}B\text{'}C\text{'}$. **

**Solution:**

Notice in this shape, the transformed shape is smaller than the original shape. However, to work out the scale factor, we do the exact same thing. We look at two corresponding sides, let's take $AB$ and $A\text{'}B\text{'}$ for example. We then divide the length of the transformed side by the length of the original side. In this case, $AB=4units$and $A\text{'}B\text{'}=2units$.

**Therefore, the scale factor, $SF=\frac{2}{4}=\frac{1}{2}$. **

Notice here that we have a **fractional** scale factor. This is always the case when we go from a **bigger** shape to a **smaller** shape.

**Below are three similar quadrilaterals. We have that $DC=10cm$, $D\text{'}C\text{'}=15cm$, $D\text{'}\text{'}C\text{'}\text{'}=20cm$and $A\text{'}D\text{'}=18cm$. Work out the area of quadrilaterals $ABCD$and $A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}D\text{'}\text{'}$. **

**Solution:**

First, let's work out the scale factor to get from $ABCD$ to $A\text{'}B\text{'}C\text{'}D\text{'}$. Since $D\text{'}C\text{'}=15cm$ and $DC=10cm$, we can say that the scale factor $SF=\frac{15}{10}=1.5$. Thus, to get from $ABCD$ to $A\text{'}B\text{'}C\text{'}D\text{'}$ we enlarge by a scale factor of $1.5$. We can therefore say that the length of $AD$ is $\frac{18}{1.5}=12cm$.

Now, let's work out the scale factor to get from $A\text{'}B\text{'}C\text{'}D\text{'}$ to $A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}D\text{'}\text{'}$. Since $D\text{'}\text{'}C\text{'}\text{'}=20cm$ and $D\text{'}C\text{'}=15cm$, we can say that the scale factor $SF=\frac{20}{15}=\frac{4}{3}$. Thus, to work out A''D'', we multiply the length of A'D' by $\frac{4}{3}$ to get $A\text{'}\text{'}D\text{'}\text{'}=18\times \frac{4}{3}=24cm$.

To work out the area of a quadrilateral, recall that we multiply the base by the height. So, the area of $ABCD$ is $10cm\times 12cm=120c{m}^{2}$ and similarly, the area of $A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}D\text{'}\text{'}$ is $20cm\times 24cm=420c{m}^{2}$.

**Below are two similar right-angled triangles $ABC$ and $A\text{'}B\text{'}C\text{'}$. Work out the length of $A\text{'}C\text{'}$. **

**Solution:**

As usual, let's start by working out the scale factor. Notice that $BC$ and $B\text{'}C\text{'}$ are two known corresponding sides so we can use them to work out the scale factor.

So, $SF=\frac{4}{2}=2$. Thus, the scale factor is $2$. Since we do not know the side $AC$, we cannot use the scale factor to work out $A\text{'}C\text{'}$. However, since we know $AB$, we can use it to work out $A\text{'}B\text{'}$.

Doing so, we have $A\text{'}B\text{'}=3\times 2=6cm$. Now we have two sides of a right-angled triangle. You may remember learning about Pythagoras' theorem. If not, perhaps review this first before continuing with this example. However, if you are familiar with Pythagoras, can you work out what we need to do now?

According to Pythagoras himself, we have that ${a}^{2}+{b}^{2}={c}^{2}$where$c$ is the hypotenuse of a right-angled triangle, and $a$ and $b$ are the other two sides. If we define $a=4cm$, $b=6cm$, and $c=A\text{'}C\text{'}$, we can use Pythagoras to work out $c$!

Doing so, we get ${c}^{2}={4}^{2}+{6}^{2}=16+36=52$. So, $c=\sqrt{52}=7.21cm$.

**We therefore have that $A\text{'}C\text{'}=7.21cm$. **

## Scale Factor Enlargement

If we have a shape and a scale factor, we can enlarge a shape to produce a transformation of the original shape. This is called an **enlargement transformation.** In this section, we will be looking at some examples relating to **enlargement transformations. **

There are a few steps involved when enlarging a shape. We first need to know **how** **much** we are enlarging the shape which is indicated by the scale factor. We also need to know **where** exactly we are enlarging the shape. This is indicated by the **centre of enlargement**.

The **centre of enlargement **is the coordinate that indicates **where** to enlarge a shape.

We use the centre of enlargement by looking at a point of the original shape and working out how far it is from the centre of enlargement. If the scale factor is two, we want the transformed shape to be twice as far from the centre of enlargement as the original shape.

We will now look at some examples to help understand the steps involved in enlarging a shape.

**Below is triangle $ABC$. Enlarge this triangle with a scale factor of $3$ with the centre of enlargement at the origin. **

**Solution:**

The first step in doing this is to make sure the centre of enlargement is labelled. Recall that the origin is the coordinate $(0,0)$. As we can see in the above image, this has been marked in as point O.

Now, pick a point on the shape. Below, I have chosen point B. To get from the centre of enlargement O to point B, we need to travel $1$ unit along and $1$ unit up. If we want to enlarge this with a scale factor of $3$, we will need to travel $3$ units along and $3$ units up from the centre of enlargement. Thus, the new point $B\text{'}$ is at the point $(3,3)$.

We can now label the point $B\text{'}$ on our diagram as shown below.

Next, we do the same with another point. I have chosen $C$. To get from the centre of enlargement O to point C, we need to travel $3$ units along and $1$ unit up. If we enlarge this by $3$, we will need to travel $3\times 3=9$ units along and $1\times 3=3$ units up. Thus, the new point $C\text{'}$ is at $(9,3)$.

We can now label the point $C\text{'}$ on our diagram as shown below.

FInally, we look at the point $A$. To get from the centre of enlargement O to the point A, we travel $1$ unit along and $4$ units up. Thus, if we enlarge this by a scale factor of $3$, we will need to travel $1\times 3=3$ units along and $4\times 3=12$ units up. Therefore, the new point $A\text{'}$ will be at the point $(3,12)$.

We can now label the point $A\text{'}$ on our diagram as shown below. If we join up the coordinates of the points we have added, we end up with the triangle $A\text{'}B\text{'}C\text{'}$. This is identical to the original triangle, the sides are just three times as big. It is in the correct place as we have enlarged it relative to the centre of enlargement.

**Therefore, we have our final triangle depicted below. **

## Negative Scale Factors

So far, we have only looked at **positive** scale factors. We have also seen some examples involving **fractional** scale factors. However, we can also have **negative** scale factors when transforming shapes. In terms of the actual enlargement, the only thing that really changes is that the shape appears to be upside down in a different position. We will see this in the below example.

**Below is quadrilateral $ABCD$. Enlarge this quadrilateral with a scale factor of $-2$ with the centre of enlargement at the point **$P=(1,1)$.** **

**Solution:**

First, we take a point on the quadrilateral. I have chosen point $D$. Now, we need to work out how far D is from the centre of enlargement P. In this case, to travel from P to D, we need to travel $1$ unit along and $1$ unit up.

If we want to enlarge this with a scale factor of $-2$, we need to travel $1\times -2=-2$ units along and $1\times -2=-2$ units up. In other words, we are moving $2$ units away and $2$ units down from P. The new point D' is therefore at $(-1,-1)$, as shown below.

Now, consider point A. To get from P to A, we travel $1$ unit along and $2$ units up. Therefore, to enlarge this with a scale factor $-2$, we travel $1\times -2=-2$ units along and $2\times -2=-4$ units up. In other words, we travel $2$ units to the left of P and $4$ units down, as shown as point A' below.

Now, consider point C. To get from P to C, we travel $3$ units along and $1$ unit up. Therefore, to enlarge this with a scale factor $-2$, we travel $3\times -2=-6$ units along and $1\times -2=-2$ units up. In other words, we travel $6$ units to the left of P and $2$ units down, as shown as point C' below.

Now, consider point B. To get from P to B, we travel $2$ units along and $2$ units up. Therefore, to enlarge this with a scale factor $-2$, we travel $2\times -2=-4$ units along and $2\times -2=-4$ units up. In other words, we travel $4$ units to the left of P and $4$ units down, as shown as point B' below.

If we join up the points, and remove the ray lines, we obtain the below quadrilateral. **This is our final enlarged shape. Notice that the new image appears upside down. **

## Scale Factors - Key takeaways

- A
**scale factor**tells us the factor by which a shape has been enlarged by. - For example, if we have a shape enlarged by a scale factor of three, then each side of the shape is multiplied by three to produce the new shape.
- The
**corresponding sides**are the sides of the shape that have proportional lengths. - If we have a shape and a scale factor, we can enlarge a shape to produce a transformation of the original shape. This is called an
**enlargement transformation.** - The
**centre of enlargement**is the coordinate that indicates**where**to enlarge a shape. - We can also have
**negative**scale factors when transforming shapes. In terms of the actual enlargement, the shape will just appear to be upside down.

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##### Frequently Asked Questions about Scale Factors

What is a scale factor?

When we enlarge a shape, the scale factor is the quantity by which each side is enlarged by.

What is a scale factor of 3?

When we enlarge a shape, we enlarge it by a scale factor of three when we multiply each of the sides by three to get the new shape.

How do you find the missing length of a scale factor?

If we know the scale factor, we can multiply the side of the original shape by the scale factor to find the missing lengths of the new shape. Alternatively, if we have known sides of the enlarged shapes, we can divide the lengths by the scale factor to get the lengths of the original shape.

How do you find the scale factor of an enlargement?

Divide the corresponding sides of the enlarged shape by the original shape.

What happens if a scale factor is negative?

The shape is turned upside down.

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