5.01 Scale factors

Corresponding angles in similar figures can be easily matched up by pairing of angles of the same size. In the case where the figures have multiple angles of the same size, we can also use an angle's neighbouring angles as another requirement to be fulfilled.

For example, in the similar figures below, both have two angles of size 98^{\circ}.

Since the neighbouring angles of \angle ABC have sizes 126^{\circ} and 117^{\circ}, we know that it must correspond with \angle PQR, and not with \angle RST.

If we know which angles are corresponding in two similar figures, then the sides opposite corresponding angles will also be corresponding.

If we do not know which angles are corresponding, we can try to determine which sides are corresponding using their relative lengths.

Since each pair of corresponding sides must have their distances in the same ratio, we can use any given side lengths to determine which side lengths must be matching.

In the case where we are given all the side lengths for both similar figures, we can match up all the sides in the figures such that they all have the same ratio between their lengths.

Examples

Example 1

These two triangles are similar.

a

Find the value of m.

Worked Solution

Create a strategy

Find the corresponding sides and equate their ratios.

Apply the idea

Side PQ corresponds to side AB, and side PR corresponds to side AC.

\displaystyle \dfrac{m}{30}	\displaystyle =	\displaystyle \dfrac{4}{12}	Equate ratios of corresponding sides
\displaystyle \dfrac{m}{30}	\displaystyle =	\displaystyle \dfrac{1}{3}	Simplify the fraction
\displaystyle m	\displaystyle =	\displaystyle \dfrac{1}{3} \times 30	Multiply both sides by 30
\displaystyle m	\displaystyle =	\displaystyle 10	Evaluate

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b

Find the value of n.

Worked Solution

Create a strategy

Find the corresponding sides and equate their ratios.

Apply the idea

Side BC corresponds to side QR, and side AC corresponds to side PR.

\displaystyle \dfrac{n}{7}	\displaystyle =	\displaystyle \dfrac{12}{4}	Equate ratios of corresponding sides
\displaystyle \dfrac{n}{7}	\displaystyle =	\displaystyle 3	Simplify the fraction
\displaystyle m	\displaystyle =	\displaystyle 3 \times 7	Multiply both sides by 7
\displaystyle m	\displaystyle =	\displaystyle 21	Evaluate

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If we are given that two figures are similar and are also given two corresponding distances, we can work out the scale factor.

The scale factor is equal to a larger distance divided by the corresponding smaller distance.

If we know the scale factor between two similar figures, then any distance in one figure can be used to calculate the corresponding distance in the other figure.

If the known distance is in the smaller figure, we can multiply by the scale factor to find the corresponding distance in the larger figure.

If the known distance is in the larger figure, we can divide by the scale factor to find the corresponding distance in the smaller figure.

In these similar figures, the scale factor is 3. Therefore, x=3\times 4=12.

Examples

Example 2

The triangles in the diagram below are similar:

-8

-6

-4

-2

2

4

6

8

x

-8

-6

-4

-2

2

4

6

8

y

What is the scale factor from \triangle ABC to \triangle A'B'C'?

Worked Solution

Create a strategy

Determine the ratio between two corresponding points' distances from the origin.

Apply the idea

Points A and A' correspond to each other.

Point A is 3 units away from the origin and point A' is 6 units away from the origin.

\displaystyle \text{Scale factor}	\displaystyle =	\displaystyle \dfrac{6}{3}	Divide 6 by 3
	\displaystyle =	\displaystyle 2	Evaluate

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Example 3

Jenny wants to find the height of her school's flag pole. During recess, she measures the length of the flag pole's shadow to be 335 \text{ cm} Her friend then measures her own shadow, which turns out to be 95 \text{ cm}.

The triangles formed by casting these shadows are similar.

If Jenny is 160 \text{ cm} tall, what is the height of the flag pole?

Round your answer to the nearest whole centimetre.

Worked Solution

Create a strategy

Equate the ratio between Jenny and the pole's heights and the ratio of the lengths of their shadows.

Apply the idea

Let h represent the height of the flag pole.

\displaystyle \frac{h}{160}	\displaystyle =	\displaystyle \frac{335}{95}	Equate the ratios of the sides
\displaystyle \frac{h}{160} \times 160	\displaystyle =	\displaystyle \frac{335}{95} \times 160	Multiply both sides 160
\displaystyle h	\displaystyle =	\displaystyle 564 \text{ cm}	Evaluate

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