10.09 Scale factors

A shape is considered an enlargement of another if each side of the first shape is multiplied by the same scale factor to form the sides of the second shape.

For example, consider a triangle with side lengths measuring 3\text{ cm},\,4\text{ cm} and 5\text{ cm}. If each side is multiplied by the same factor, say 2, the new resulting triangle will have side lengths measuring 6\text{ cm},\,8\text{ cm} and 10\text{ cm}. The resulting shape is larger.

Examples

Example 1

Which two of these shapes are enlargements of each other?

A

B

C

D

Worked Solution

Create a strategy

Choose options that have same shape but with different sizes.

Apply the idea

The two triangles that are enlargements of each other are in option A and option D.

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Example 2

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

Worked Solution

Create a strategy

Divide the side length of the larger shape by the corresponding side length of the smaller shape.

Apply the idea

\displaystyle \text{Scale Factor}	\displaystyle =	\displaystyle \dfrac{A'D'}{AD}	Divide side A'D' by side AD
	\displaystyle =	\displaystyle \dfrac{12}{4}	Substitute A'D'=12 and AD=4
	\displaystyle =	\displaystyle 3	Evaluate

\displaystyle \text{Scale Factor}	\displaystyle =	\displaystyle \dfrac{D'C'}{DC}	Divide side D'C' by side DC
	\displaystyle =	\displaystyle \dfrac{6}{2}	Substitute D'C'=6 and DC=2
	\displaystyle =	\displaystyle 3	Evaluate

\displaystyle \text{Scale Factor}	\displaystyle =	\displaystyle \dfrac{C'B'}{CB}	Divide side C'B' by side CB
	\displaystyle =	\displaystyle \dfrac{3}{1}	Substitute C'B'=3 and CB=1
	\displaystyle =	\displaystyle 3	Evaluate

Shape A' B' C' D' has all side lengths 3 times larger than the corresponding sides of shape ABCD so the shapes are similar with a scale factor of 3.

A shape is considered a reduction of another if each side of the first shape is divided by the same scale factor to form the sides of the second shape.

Consider the reverse of the above example-a triangle with side lengths measuring 6 \text{ cm},\, 8 \text{ cm} and 10 \text{ cm} has each side multiplied by a factor of \dfrac{1}{2}. The new resulting triangle will have side lengths measuring 3 \text{ cm},\, 4 \text{ cm} and 5 \text{ cm}. The resulting shape is smaller than the original.

The scale factor tells us by how much the object has been enlarged or reduced.

\text{linear scale factor, k}=\frac{\text{length of image}}{\text{length of object}}

If the scale factor is greater than 1, then the image is bigger than the original.

If the scale factor is less than 1, then the image is smaller than the original.

Examples

Example 3

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

A

\dfrac{1}{4}

B

3

C

\dfrac{1}{3}

D

4

Worked Solution

Create a strategy

We need to find the number that can multiply to the side length of the bigger triangle that is equal the side length of the small triangle.

Apply the idea

We are given: B'O = 20

Option A:

\displaystyle 15 \times \dfrac{1}{4}

\displaystyle =

\displaystyle 3.75

Evaluate the multiplication

Option B:

\displaystyle 15 \times 3

\displaystyle =

\displaystyle 45

Evaluate the multiplication

Option C:

\displaystyle 15 \times \dfrac{1}{3}

\displaystyle =

\displaystyle 5

Evaluate the multiplication

Option D:

\displaystyle 15 \times 4

\displaystyle =

\displaystyle 60

Evaluate the multiplication

Side B to O have a side of 5.

Since 15 is multiply by \dfrac{1}{3} we can get 5, so the correct answer is option C.

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Suppose we have a square with side lengths 2\text{ cm} and we enlarge each side length by a scale factor of 3.

Lengths have been scaled by a factor 3, but our area has gone from 4 \text{ cm}^2 to 36 \text{ cm}^2. The ares has been scaled by a factor of 9.

Areas of similar figures do not scale by the same factor as the the scale factor of the length, and for this reason, they have their own scale factor called the area scale factor.

If enlarging or shrinking any figure, no matter how irregular, by a length scale factor of k, the areas of all these squares inside it will each scale by a factor of k^{2}.

For any figure that is scaled by a length scale factor of k, the area scale factor will be k^{2}.

Examples

Example 4

In each of the following cases:

a

Find the value of x.

Worked Solution

Create a strategy

We need to find first the width of the two rectangles and then find the value of x by using the formula: \text{Area of rectangle}=l \times w.

Apply the idea

The width of the large rectangle:

\displaystyle 324	\displaystyle =	\displaystyle 27 \times w	Substitute A=324 and l=27
\displaystyle w	\displaystyle =	\displaystyle \dfrac{324}{27}	Divide both sides by 27
	\displaystyle =	\displaystyle 12\text{ cm}	Evaluate

The width of the small rectangle:

\displaystyle w	\displaystyle =	\displaystyle 12\div (27\div9)	Divide the lengths of the triangles
	\displaystyle =	\displaystyle 12\div 3	Evaluate the division inside the brackets
	\displaystyle =	\displaystyle 4\text{ cm}	Evaluate

The value of x:

\displaystyle A	\displaystyle =	\displaystyle l \times w	Use the formula
\displaystyle x	\displaystyle =	\displaystyle 9 \times 4	Substitute A=x,\,l=9 and w=4
	\displaystyle =	\displaystyle 36\text{ cm}^2	Evaluate

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b

Find the value of x.

Worked Solution

Create a strategy

Find first the scale factor, then squared it and then multiply it by the area of the small hexagon.

Apply the idea

Since the two hexagon are similar, which mean all the pair of corresponding sides are same proportion.

\displaystyle \text{Scale factor}	\displaystyle =	\displaystyle 21 \div 7	Divide the side lengths
	\displaystyle =	\displaystyle 3\text{ cm}	Evaluate

To find the x:

\displaystyle x	\displaystyle =	\displaystyle 3^{2} \times 56	Squared the \text{Scale factor}=3 and mltiply by \text{Area}=56
	\displaystyle =	\displaystyle 504\text{ cm}^2	Evaluate

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Given a cube of side length a units, its volume would be a^3 \text{ units}^3.

If scaling any figure, no matter how irregular, by a length scale factor of k, the volumes of all these cubes inside it will each scale by a factor of k^3.

Examples

Example 5

Consider the two similar trapezoidal prisms.

a

Find the length scale factor, going from the smaller prism to the larger prism.

Worked Solution

Create a strategy

Divide the larger base by the corresponding smaller base.

Apply the idea

\displaystyle \text{Length scale factor}	\displaystyle =	\displaystyle \dfrac{8}{4}	Divide the bases
	\displaystyle =	\displaystyle 2	Evaluate

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b

Find the volume scale factor, going from the smaller prism to the larger prism.

Worked Solution

Create a strategy

Use the formula: \text{Volume scale factor} = \text{(Length scale factor)}^3

Apply the idea

\displaystyle \text{Volume scale factor}	\displaystyle =	\displaystyle 2^3	Cube the length scale factor
	\displaystyle =	\displaystyle 8	Evaluate

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