Scale Factors

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

Working out scale factor example with quadrilaterals - StudySmarter Originals

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 $A\text{'}D\text{'}$.

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 $A\text{'}B\text{'}=ycm$.

AB=4 cmWork out the value of $x$ and $y$.

Example working out missing lengths using scale factor - StudySmarter Originals

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 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 $A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}D\text{'}\text{'}$.

Example working out the area using scale factor - StudySmarter Originals

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 $A\text{'}C\text{'}$.

Working out missing length using scale factor and pythagoras - StudySmarter Originals

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 $A\text{'}C\text{'}=7.21cm$.

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

Example of enlarging a triangle - StudySmarter Originals

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 $(3,3)$.

Example of enlarging a triangle - StudySmarter Originals

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

Example of enlarging a triangle point by point - StudySmarter Originals

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 $(9,3)$.

Example of enlarging a triangle point by point - StudySmarter Originals

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

Example of enlarging a triangle point by point - StudySmarter Originals

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 $(3,12)$.

Example of enlarging a triangle point by point - StudySmarter Originals

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.

Example of enlarging a triangle - StudySmarter Originals

Therefore, we have our final triangle depicted below.

Example of enlarging a triangle - StudySmarter Originals

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

Negative scale factors example - StudySmarter Originals

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.

Negative scale factors example - StudySmarter Originals

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.

Negative scale factors example - StudySmarter Originals

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.

Negative scale factors example - StudySmarter Originals

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.

Negative scale factors example - StudySmarter Originals

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.

Negative scale factors example - StudySmarter Originals