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.

Worked example

example 1

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.

Solve for $x$x.

Solve for $c$c.

Question 3

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.

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

Question 4

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