# 4.04 Lengths of Similar figures

Lesson

### Similar figures

Two shapes, or figures, are said to be similar if they are the same shape but different sizes. Two key features of similar figures are:

• Corresponding angles are equal
• Corresponding sides are proportional, meaning they have been multiplied by the same scale factor.

Similar shapes are either enlargements or reductions of one another.

### Enlargements

A shape is considered an enlargement of another if one shape has side lengths that are all increased by the same scale factor.

#### Worked example

Take a triangle with side lengths measuring $3$3 cm, $4$4 cm and $5$5 cm. If each side is multiplied by the same factor, say $2$2, the new resulting triangle will have side lengths measuring $6$6 cm, $8$8 cm and $10$10 cm. 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$6 cm, $8$8 cm and $10$10 cm has each side multiplied by a factor of $\frac{1}{2}$12. The new resulting triangle will have side lengths measuring $3$3 cm, $4$4 cm and $5$5 cm. The resulting shape is smaller than the original.

Scale Factor

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

The scale factor can be greater than $1$1, so the image is being made bigger than the original.

The scale factor can be smaller than $1$1, so the image is being made smaller than the original.

#### Worked example

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

To find the scale factor we:

a) identify corresponding sides, in some cases this might mean rotating the shape.

b) compare the ratios of the matching sides.

By aligning the largest length sides, AD and A'D', with each other then using this to match up the other sides, we can produce 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

#### Worked example

The following triangles are similar shapes. Find the scale factor and the value of $u$u.

Think: Compare the size of the matching sides to find the scale factor.

Do:

 scale factor $=$= $\frac{3}{21}$321​ $=$= $\frac{1}{7}$17​ the triangle with $u$u is smaller, so it makes sense that the scale factor is less than $1$1.

Think: Compare the size of the matching sides to find the scale factor.

Do:

 $u$u $=$= $\frac{1}{7}\times14$17​×14 $=$= $2$2

#### Practice questions

##### Question 1

In the figure below the scale factor used to enlarge the smaller quadrilateral is $4.5$4.5. Find the length of $FG$FG.

##### Question 2

Consider the two similar triangles.

1. Solve for $x$x.

2. Solve for $c$c.

##### Question 3

The green rectangle is an enlargement of the blue rectangle.

1. What is the enlargement factor?

2. Write the scale of the blue rectangle to the green rectangle.

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### Problem solving using similar shapes

The sides lengths of similar shapes are in the same ratio or proportion. So once we know that two shapes are similar, we can find any unknown side lengths by using the ratio.

Remember!

We can write the ratio of the big triangle to the little triangle or the little triangle to the big triangle. This is helpful as it means we can always have the unknown variable as the numerator.

#### Practice questions

##### Question 4

A stick of height $1.1$1.1 m casts a shadow of length $2.2$2.2 m. At the same time, a tree casts a shadow of $6.2$6.2 m.

1. If the tree has a height of $h$h metres, solve for $h$h.

##### Question 5

A $4.9$4.9 m high flagpole casts a shadow of $4.5$4.5 m. At the same time, the shadow of a nearby building falls at the same point (S). The shadow cast by the building measures $13.5$13.5 m. Find $h$h, the height of the building, using a proportion statement.

### Outcomes

#### MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures