Mathematics Stage 5
SGS5.2.2
Part 3
Properties of geometrical figures
Similar triangles
Number: 43690
Title: SGS.2.2 Properties of Geometrical Figures
This publication is copyright New South Wales Department of Education and Training (DET), however it may contain
material from other sources which is not owned by DET. We would like to acknowledge the following people and
organisations whose material has been used:
Outcomes from Mathematics Years 7-10 Syllabus © Board of Studies, NSW 2002
www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf
Overview, pp iii-iv
RSL Memorial photo © Thomas Brown. Reproduced with permission.
Part 1, p 17
Various Wingeom screenshots © Rick Parris/Wingeom. Reproduced with permission.
P1, 2 & 3, various
pages.
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Contents – Part 3
Introduction – Part 3 ..........................................................3
Indicators ...................................................................................3
Preliminary quiz.................................................................5
Similar figures ...................................................................8
Scale factors ...................................................................15
Ratios of corresponding sides .........................................21
Writing equations.............................................................27
How high is that?.............................................................33
Suggested answers – Part 3 ...........................................39
Additional resources – Part 3 ..........................................50
Exercises – Part 3 ...........................................................54
Part 3
Similar triangles
1
2
SGS5.2.2 Properties geometrical figures
Introduction – Part 3
In this part you will investigate the relationship between pairs of
matching sides of similar figures and in particular, similar triangles.
You will use this relationship to calculate unknown sides. You will also
revise the fact that matching angles in similar figures are always equal
and then use these two properties to identify when figures are similar.
Indicators
By the end of Part 3, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
•
applying the enlargement or reduction factor to find unknown sides
in similar triangles
•
identifying the elements preserved in similar triangles, namely angle
size and the ratio of corresponding sides
•
determining whether triangles are similar
•
calculating unknown sides in a pair of similar triangles.
By the end of Part 3, you will have been given the opportunity to work
mathematically by:
•
explaining why any two equilateral triangles, or any two squares
are similar
•
investigating whether any two rectangles, or any two isosceles
triangles are similar
•
Part 3
Using dynamic geometry software to investigate the properties of
similar triangles
Similar triangles
3
•
applying the properties of similar triangles to solve problems,
justifying the results.
Source:
4
Adapted from outcomes of the Mathematics Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_
710_syllabus.pdf > (accessed 04 November 2003).
© Board of Studies NSW, 2002.
SGS5.2.2 Properties geometrical figures
Preliminary quiz
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1
a
Show that
1
×16 gives the same answer as 16 ÷ 2 ?
2
___________________________________________________
b
2
3
Complete: 45 ÷10 = ___× 45
The inverse of
3
4
is . What is the inverse of
4
3
a
1
?_________________________________________________
2
b
3?__________________________________________________
a
What is the highest common factor of 12 and 18? __________
b
Reduce
c
i
What number would you use to cancel down in one step,
75
, to lowest form? _____________________________
100
ii
What is the relationship between this number, 75 and 100?
12
to lowest form. ____________________________
18
________________________________________________
Part 3
Similar triangles
5
4
a
Write down the keys you must press on your calculator to find
1
6 × 47 . Calculate the answer.
2
___________________________________________________
b
What decimal must you multiply 47 by to get the same answer?
___________________________________________________
5
a
Simplify the ratio (pronounced ‘raysheeoh’) 18:8 ___________
Note: you say 18:8 like this: ‘eighteen to eight’.
b
Complete the following by writing a different number in
each square.
6
i
The ratio 3:5 may be written in the form
ii
3:5 is equivalent to
iii
3
=
5
.
:1.
35
Solve each equation, using a calculator to get the final answer.
a
x 5
=
41 8
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
24x = 30
___________________________________________________
___________________________________________________
___________________________________________________
6
SGS5.2.2 Properties geometrical figures
7
Find x o in the diagram below. Give a reason for your answer.
42°
x°
_______________________________________________________
8
Circle the correct words below this diagram.
50°
40°
50°
40°
The triangles above are similar/congruent because their matching
angles are equal.
Check your response by going to the suggested answers section.
Part 3
Similar triangles
7
8
SGS5.2.2 Properties geometrical figures
Similar figures
Two figures are similar if you can rotate, reflect and/or slide one figure
and also enlarge or reduce it (make it smaller) so it sits directly on top of
the other.
In the diagram below, the original figure is reflected to figure A.
After this it slides to figure B, is reduced to figure C, rotated about O to
figure D and then finally enlarged. The original and final figures are
similar (and so are all the ones in between).
B
slide
C
O
D
final
figure
original
figure
A
reflect
Can you describe the transformations
that the final figure goes through in
order to get back to the original?
Try to describe it yourself first. The answer is below.
Start with the final figure ¥ Reduce to figure D ¥Rotate to figure C
¥Enlarge to figure B ¥ Slide to figure A ¥ Reflect to original figure.
The website below is another animation of similar triangles.
Can you describe the transformation as one triangle is placed directly on
top of the other?
Part 3
Similar triangles
9
Access this site on similar triangles by visiting the CLI webpage below.
Select Mathematics then Stage 5.2 and follow the links to resources for
this unit SGS5.2.2 Properties of geometrical figures, Part 3.
<http://www.cli.nsw.edu.au/Kto12>
Reflections (flips), rotations (spins) and translations (slides) produce only
congruent figures. A congruent figure is just one of a group of
similar figures.
Enlargements and reductions produce all other similar figures.
These are created by using a scale factor. For example if a figure is
three times as big as another then the scale factor = 3.
Use the activity below to review work studied previously about similar
triangles and scale factors.
10
SGS5.2.2 Properties geometrical figures
Activity – Similar figures
Try these.
1
If this triangle is doubled in size,
4
47°
cm
3c
m
29°
104°
2 cm
write the new measurements on its enlargement below.
Part 3
Similar triangles
11
2
The diagram below shows two triangles that have their matching
vertices joined by straight construction lines. These lines meet at
one point.
10
m
m
C
m
5m
B
A
a
b
c
What is the scale factor used if the:
i
smaller triangle is enlarged (using the construction lines,) to
sit directly on top of the large triangle? _______________
ii
larger triangle is reduced (using the construction lines,) to
sit directly on top of the small triangle? _______________
If AB = 20 mm, how long is:
i
BC _____________________________________________
ii
AC _____________________________________________
Complete using the diagram above:
i
Enlargement: The scale factor of 2 is the same as
5
ii
12
or
.
Reduction: The scale factor of ___ is the same as
5
40
40
or
.
SGS5.2.2 Properties geometrical figures
Check your response by going to the suggested answers section.
If one figure is an enlargement (or a reduction) of another, then the two
figures are similar. If you double the side length of a figure, the scale
1
factor = 2. If you halve the side length of a figure, the scale factor = .
2
Similarly, if you reduce the side length by dividing it by five, then the
1
scale factor = .
5
In all cases the angles remain the same but the sides of the figure get
enlarged or reduced by this same factor.
You have been reviewing simple enlargements and reductions.
Now check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.1 – Similar figures.
Part 3
Similar triangles
13
14
SGS5.2.2 Properties geometrical figures
Scale factors
When you enlarge (or reduce) a figure, its lengths change but the angles
remain the same. The scale factor used determines the size of the new figure.
What factors enlarge a shape?
Factors like 2, 3 and 4.
Or 2.6, 3
1
or 4.12345.
2
What about 1.5? Does that
make it bigger or smaller?
Bigger, you must times every length by
1.5. This will make the shape bigger.
Susana uses the diagram and calculation below to explain.
m
m
15
10
m
m
This means that the matching
side of the new triangle is
15 mm, so it is bigger.
10 × 1.5 = 15
The class conclude that all the factors bigger than one enlarge a shape.
Part 3
Similar triangles
15
The class now use their calculators to find out if the factors below
enlarge or reduce ten. Do these calculations yourself. Is the answer
bigger or smaller than ten in all cases?
1
2
1
10 ×
3
10 × 0.1
3
10 ×
4
10 ×
These answers are all less than ten. The class now realise that factors
1 1
3
like , , 0.1 and
reduce a shape. This is because all of these factors
2 3
4
are between zero and one. When you multiply any number by the
numbers between zero and one, you get a smaller number, so the new
lengths will be smaller.
Does this mean that a scale
factor of one does nothing.
Yes. A scale factor of one just
duplicates the picture. The two
figures would be congruent.
The example below demonstrates how scale factors affect shapes.
It also introduces a new word that will replace the word ‘matching’.
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
One side of a figure is 6 mm long. This figure is enlarged
by a scale factor of 4.2. How long is the correspnding
side (matching side) of the larger shape?
b
The length of one side of a figure is 120 mm and the
corresponding side of a similar figure is 12 mm.
What scale factor was used to reduce the larger figure to
the smaller figure?
16
SGS5.2.2 Properties geometrical figures
Solution
a
A scale factor of 4.2 means that you must multiply every
length of the new figure by 4.2.
Corresponding length=6 × 4.2
=25.2 mm
The following two dot points are about the word
‘corresponding’:
•
The word ‘corresponding’ means ‘same position’.
In similar triangles the sides that are in the same
position are corresponding sides. (You can also call
them matching sides.)
•
The word ‘corresponding’ has the same meaning in the
diagram below.
transversal
corresponding angles
These angles are in the same position. They are both
below a parallel line and to the right of the transversal.
Part 3
Similar triangles
17
b
120 mm has been reduced to 12 mm, so
120 × scale factor = 12 .
There are many ways you can find this answer.
Method 1
120 × scale factor=12 , so dividing both sides by 120.
120 × scale factor 12
=
120
120
∴scale factor=
12
120
12
can be reduced to a simpler form by dividing 12 and
120
120 by their highest common factor of 12.
12
Scale factor =
120
1
=
10
(Note: you could also have calculated 12 ÷120 = 0.1 )
Method 2
You can see that 120 ÷10 = 12 . Remember that dividing
by ten is exactly the same as multiplying by one tenth.
So 120 ÷10 = 12 is the same as 120 ×
∴ The scale factor =
18
1
= 12 .
10
1
10
SGS5.2.2 Properties geometrical figures
Method 3
This is a reduction, so the scale factor must be less than
one, so put the smaller number on the top of the fraction
and the larger number on the bottom.
small number
large number
12
=
120
1
=
10
Scale factor=
The quickest method is the last method, but it is more intuitive which
means that you must understand what you are doing. You must first
decide if it is an enlargement or a reduction. If it is a reduction, then the
scale factor is less than one, so you must divide the smaller number by
the bigger number. If it is an enlargement, you do the reverse.
This information is summarised in the table below.
Transformation
Size of scale factor
Calculation
Reduction:
scale factor
small number
large number
between 0 and 1
Enlargement:
scale factor > 1
big number
small number
You have been practising using scale factors. Now check that you can
solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.2 – Scale factors.
Part 3
Similar triangles
19
20
SGS5.2.2 Properties geometrical figures
Ratios of corresponding sides
The similar triangles below:
•
have equal corresponding angles
•
are enlargements or reductions of each other
•
have scale factors that determine their size.
vanishing point
Scale factor =
3
or 0.6
5
6
0.
1
m
m
original
triangle
You will investigate these triangles further in the activity below.
Part 3
Similar triangles
21
Activity – Ratios of corresponding sides
Access Wingeom by visiting the CLI webpage below. Select
Mathematics then Stage 5.2 and follow the links to resources for this unit
SGS5.2.2 Properties of geometrical figures, Part 3
<http://www.cli.nsw.edu.au/Kto12>
Try these.
Step1
Open Wingeom, following the steps in the additional resources.
Step 2
Highlight the file called ‘Ratios of corresponding sides’
Your window should look like this.
Source
© Wingeom
Step 3
Read all the ratios on the window and look at the diagram to see
where they match the diagram.
(Note: DE/AB is the way this program writes ratios, so DE/AB
DE
represents
, DE:AB or DE ÷ AB)
AB
22
SGS5.2.2 Properties geometrical figures
Step 4
Follow the instructions written in red.
1
a
What is the ratio when the two triangles sit directly on top of
each other? __________________________________________
b
V is called the vanishing point. Why?
___________________________________________________
c
What is the ratio when the moving triangle is at the vanishing
point? ______________________________________________
d
Complete:
i
All three ratios of corresponding sides are ______________
ii
All three ratios of ________________ from the
vanishing point are ______________.
iii Another name for the “current value of #” is ____________
_____________.
Step 5
Close Wingeom.
Check your response by going to the suggested answers section.
When one triangle is reduced (or enlarged,) then the scale factor is the
same as:
•
all three ratios of corresponding sides
•
all three ratios of each triangle’s corresponding distance from the
vanishing point.
These things are only equal if you compare corresponding lengths in the
1
correct order. Remember, order is important because is a reduction
2
2
and
is an enlargement.
1
Note that 1:2 is the inverse of 2:1. (Remember that if you turn a fraction
upside down, you get its inverse.)
2 1
=
1 2
∴ inverse of 2:1=1:2
Inverse of
Part 3
Similar triangles
23
In the activity below you will use these facts to decide if certain figures
are similar.
Activity – Ratios of corresponding sides
Try these.
2
a
Calculate the following ratios.
A
D
m
7
m
F
B
54 mm
b
m
m
m
36
m
m
6m
42
9 mm
E
C
i
AB
____________________________________________
DE
ii
BC:EF __________________________________________
Provide working that shows that the last pair of corresponding
sides have the same ratio.
___________________________________________________
c
What do these ratios tell you about the pair of triangles?
___________________________________________________
d
Complete:
the ratio, DF:AC = _____:_____. This ratio is the __________
of 6:1.
___________________________________________________
24
SGS5.2.2 Properties geometrical figures
3
Answer questions about the rectangles below.
10 cm
B
E
7 cm
F
4 cm
2.8 cm
A
D
C
a
H
G
Use AB and EF to calculate the scale factor that was used to
reduce 10 cm to 7 cm.
___________________________________________________
___________________________________________________
b
Show that the other pair of corresponding sides have the same
scale factor.
___________________________________________________
___________________________________________________
c
Complete this sentence:
These rectangles are similar because the ratios of ____________
___________________________________________________
Check your response by going to the suggested answers section.
Two triangles are similar if:
•
two pairs of angles are equal. (Remember that this means that the
third pair of angles must also be equal)
•
the ratios of all corresponding sides are equal.
There are other ways of testing if triangles are similar. These tests are
related to the congruent triangles that you studied previously, but it is not
necessary for you to know about these. If you are interested in finding
out about other tests that you can use to decide if two triangles are
similar, you can go to the CLI website below to learn more.
Part 3
Similar triangles
25
Access the PDF file on tests for similar triangles by visiting the CLI
webpage below. Select Mathematics then Stage 5.2 and follow the links to
resources for this unit SGS5.2.2 Properties of geometrical figures, Part 3.
<http://www.cli.nsw.edu.au/Kto12>
You have been practising calculating the ratios of corresponding sides.
Now check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.3 – Ratios of
corresponding sides.
26
SGS5.2.2 Properties geometrical figures
Writing equations
You can calculate the lengths of sides in similar triangles because the
ratios of their corresponding sides are the same. Information from one
figure can help you find unknown lengths of another.
To do this you need to make equations using these ratios. You must
always make sure that you write these pairs of ratios in the correct order.
If you don’t, you will write the wrong equation!
Equations of corresponding ratios are used in the following example.
Follow through the steps in this example. Do your own working in the
margin if you wish.
The following triangles are similar because two pairs of
corresponding angles are equal.
D
A
B
E
C
F
a
The equations below all start with AB. Circle all the correct
equations.
AB BC
=
DE EF
b
AB EF
=
DE BC
AB DE
=
BC EF
AB EF
=
BC DE
If DE = 8.4 m, EF = 7.2 m and BC = 6 m, calculate AB
using one of the correct equations above.
Part 3
Similar triangles
27
Solution
a
AB BC
=
DE EF
AB EF
=
DE BC
AB BE
=
BC EF
AB EF
=
BC DE
The first equation is correct not only because corresponding
sides are matched but also because the shapes are written in
the correct order. This is because
length of 1st shape
AB
and
=
DE corresponding length of 2nd shape
other length of 1st shape
BC
=
EF other corresponding length of 2nd shape
These are equivalent ratios.
The third equation is also correct because corresponding
sides are matched across the equal sign and the correct
shapes have been kept together on the same side of the equal
sign. This is because
length of 1st shape
AB
and
=
BC other length of 1st shape
corresponding length of 2nd shape
DE
=
EF other corresponding length of 2nd shape
These are also equivalent ratios.
The second equation is wrong because, although it has used
corresponding sides, the shapes are in different orders. This
is because:
length of 1st shape
AB
and
=
DE corresponding length of 2nd shape
other length of 2nd shape
EF
=
BC corresponding length of 1st shape
These ratios are not equal. One of the ratios is the inverse of
1 2
the other. It is the same as saying, for example, that = .
2 1
This is obviously wrong.
Similarly, the last equation is incorrect. The corresponding
sides do not match across the equal sign.
28
SGS5.2.2 Properties geometrical figures
b
AB BC
AB DE
and
will both give the same answer.
=
=
DE EF
BC EF
Substituting into the first equation:
6
AB
=
8.4 7.2
Now multiply both sides by 8.4
8.4 ×
AB
6
=
× 8.4
8.4 7.2
This means that
AB =
6
× 8.4
7.2
Now you can use your calculator,
AB = 6 ÷ 7.2 × 8.4
AB = 7 m
Notice that the unknown (AB) was purposefully put in the top left
position of the equation. This makes it easier to solve the equation.
7.2 8.4
is quite correct but more difficult to solve
=
AB
6
because the unknown is on the bottom of the fraction. Avoid writing
This equation,
your equations like this.
Lastly, choose the equation you feel happier writing. Take time to write
your equations, then recheck them. There is no point solving an
incorrect equation.
Practise this in the following activity.
Part 3
Similar triangles
29
Activity – Writing equations
Try these.
1
Answer the following questions about the similar figures below.
A
D
20 cm
12 cm
C
m
0c
5
F
x
E
B
a
Complete this equation using the sides AB and DE.
EF
=
BC
b
Substitute the correct values into this equation by completing the
following.
x
=
___________________________________________________
c
Solve this equation to find x.
___________________________________________________
___________________________________________________
___________________________________________________
30
SGS5.2.2 Properties geometrical figures
2
In the diagram below, the big triangle ABC is similar to the smaller
triangle ADE.
A
10 m
m
x
E
m
C
7m
D
12
mm
B
a
Redraw the two triangles as separate diagrams in the space
below. Write the correct measurements along the correct sides.
b
Write a correct equation that uses the ratios of corresponding
sides of these similar triangles.
___________________________________________________
c
Solve this equation to find x.
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Part 3
Similar triangles
31
Writing ratios in order is absolutely important when you are writing
1 2
equations. When using just one ratio, say, 1:2 = 1:2, you can write =
1 2
1 1
2 2
1 2
and = and = but you must remember that ≠ and vice
2 2
1 1
2 1
versa.
You have been practising writing equations. Now check that you can
solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.4 – Writing equations.
32
SGS5.2.2 Properties geometrical figures
How high is that?
Writing equations using the ratios of corresponding sides in similar
triangles has a practical use.
People have known about this practical use since ancient times.
2500 years ago Thales, a Greek philosopher and engineer used similar
triangles to calculate the height of the great pyramid in Egypt and
probably other tall objects.
Thales measured the shadow that the pyramid cast on a sunny day.
He possibly may have used the 45 0 right-angled isosceles triangle to
calculate the height because in this triangle the height of the object would
be the same as the length of the shadow, or he may have used similar
triangles. This difference in his calculation, however, is
purely conjecture.
You are going to use similar triangles and shadows to measure the height
of something tall.
Part 3
Similar triangles
33
Activity – How high is that?
Try these.
Choose something tall, such as your house, goal posts, a tree or a shed.
(Mountains could be a bit difficult because you will need to know how
far the base of the mountain is from the edge of its shadow!)
You will need a sunny day, a ruler and pencil, something to mark the
ground with, such as chalk, and the table and diagram provided for you at
the end of this practical task. You may use a calculator.
Before measuring any heights, you need to measure your average pace.
Step 1
Make a mark on the ground where you are standing and take one
step. Measure the length of this step using your ruler and record
this length in your table.
Step 2
Repeat this process two more times.
Step 3
Add the lengths of all three paces. Record this in the table.
Step 4
Divide the total length of the first three paces by three to find
your average pace length. Record this average in the table.
If you had a long tape measure it would be better to take ten steps,
measure this length and divide by ten to find your average pace, but this
is too difficult if you only have a ruler.
Step 5
34
Decide which tall object you are going to measure and find its
shadow. (This is an important decision. Choose an object on
flat ground, where you can easily measure from its base.
This will make your measurements more accurate.)
SGS5.2.2 Properties geometrical figures
Step 6
Mark any point on the ground where the object’s shadow
corresponds to the tallest part of the object.
shadow
mark the first
point here
Step 7
Make sure your back is to the Sun and walk backwards until
your head just comes inside the shadow of the object.
(If you bump into the object, either the object is not tall enough
or you need to wait for its shadow to get longer.)
Step 8
Mark the point just in front of your toes.
Step 9
Use a ruler to measure the distance between these two points.
Record this distance in the table.
If it is a large distance, step out the distance between the two
points and calculate this distance by multiplying the number of
paces by your average pace length that you calculated previously.
Step 10 Step out the distance between your first point at the edge of the
shadow and the base of your object. Write this number of paces
next to your table. Multiply this number of paces by your
average pace length and record your answer in the table.
Step 11 Measure your own height and record this in the table.
Step 12 Write the appropriate measurements in the correct place on the
diagram provided for you below the table.
Step 13 Write an equation under the diagram that uses the ratios of
corresponding sides of this pair of similar triangles. Make sure
you write the equation, so that the unknown height is on the top
left of your equation.
Part 3
Similar triangles
35
Step 14 Calculate the height of the object.
1
Length of pace 1
Length of pace 2
Length of pace 3
Total length of first three paces
Your average pace length
Distance between first and second point on the ground
Distance between edge of shadow and base of object
unknown height
of object
My height
your
height
distance between first
and second mark
distance between first point and base of object
Write your own measurements on the diagram below.
Write your equation below, then solve it to find the unknown height.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
36
SGS5.2.2 Properties geometrical figures
You have been using your own measurements and similar triangles to
calculate the height of a tall object. Copy these results for your teacher
in the spaces provided for you in the following exercise.
Go to the exercises section and complete Exercise 3.5 – How high is that?
Part 3
Similar triangles
37
38
SGS5.2.2 Properties geometrical figures
Suggested answers – Part 3
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1
a
1 × 168= 8
12
(To cancel, you must divide both 16 and 2 by 2.)
Also 16 ÷ 2 = 8
Why are they the same?
Because
1
×16 = 1×16 ÷ 2
2
= 16 ÷ 2
1
by 16 you are actually
2
multiplying 16 by 1 (= 16) and then dividing 16 by 2.
Although you are multiplying
b
1
10
Similarly
2
Part 3
1
× 45 = 1× 45 ÷10
10
= 45 ÷10
a
1
2
= 2 Simply turn your half upside down, like this 2
1
b
3
1
. Remember 3 =
Now turn it upside down.
1
3
Similar triangles
39
3
a
6. The factors of 12 = 1, 2, 3, 4, 6, 12
The factors of 19 = 1, 2, 3, 6, 9, 18
The common factors of 12 and 18 are 1, 2, 3, 6 . These are the
factors that are the same in both lists.
The highest number in this last list is 6. This is why 6 is called
the highest common factor of 12 and 18.
b
2
3
You can do this the slow way, by dividing by some of the lower
common factors, like this:
÷3
÷2
12
6
2
=
=
18
9
3
÷2
÷3
or you can do this the fast way by dividing both numbers by six
immediately, like this:
÷6
12 2
=
18
3
÷6
c
i
25 ( 75 ÷ 25 = 3, 100 ÷ 25=4 )
ii
25 is the highest common factor of 75 and 100
It is the biggest number that can divide exactly into both
75 and 100 at the same time.
4
ab⁄c
a
6
b
6.5
1
ab⁄c
2 × 47 = 305
1
2
1
6
3
6.5 is the same as 6 . (Similarly 6.6 = 6 = 6 )
2
10
5
40
SGS5.2.2 Properties geometrical figures
5
a
18:8 = 9:4 (By dividing both numbers by 2.)
Alternatively, you can think of ratios as fractions.
18 9
=
4
8
Note: you may also think of ratios as a division, like this.
18 ÷ 8 = 2.25 , because 18:8 = 2.25:1.
b
i
3
5
ii
0.6 (3 ÷ 5 = 0.6)
The ratio symbol can mean different things in different
situations. The symbol ‘:’ can have the same meaning as
the line used between the top and bottom number of a
fraction. It also can have the same meaning as ÷ .
Notice that the symbol for division, ‘ ÷ ’ is a combination of
these two symbols.
In many parts of the world ‘:’ is used instead of ÷ .
So for example, when they write 3:5, they mean 3 ÷ 5.
iii 21
You need to ask yourself, 5 × ? = 35 . ? = 7.
×?
3
=
5 35
×7
Now calculate 3 × 7 .
Alternatively, you could use algebra to solve it.
First multiply both sides of the equation by 35.
35
35
Part 3
Similar triangles
×
3
35
=
×
5 35
×
3
=
5
41
Then cancel 35 and 5 by dividing by 5.
35
7
3
× =
5
1
7 × 3 = 21
6
a
25.625
Multiply both sides by 41 and cancel, to remove 41 from the left
hand side of the equation.
41×
b
x 5
= × 41
41 8
5
x = × 41
8
x = 5 × 41÷ 8
= 25.625
1.25
Divide both sides of the equation by 24 and cancel to remove 24
from the left hand side of the equation.
24x 30
=
24
24
30
24
30
= 30 ÷ 24 (Remember
is the same as 30 ÷ 24 .)
24
= 1.25
x=
7
x o = 42 o (corresponding angles are equal in // lines.)
Note: the symbol // can be replaced by ‘parallel’ or ‘is parallel to’
depending on the context.
For example AB//CD means AB is parallel to CD.
B
A
D
C
42
SGS5.2.2 Properties geometrical figures
8
Similar.
If you wish to revisit this concept and investigate this diagram more
thoroughly, you can do this by using the geometry software package
called Wingeom, below.
You will need a Windows®-based computer and the special software
package called Wingeom. You can download Wingeom from the link for
this part on the CLI website.
<http://www.cli.nsw.edu.au/Kto12>
Step 1
Open Wingeom, following the first three steps provided for you
in the additional resources.
Step 2
Highlight the file called ‘Similar Triangles’ and click on ‘open’.
Your screen should look like this.
Source
© Wingeom
Step 3
Follow the instructions on the window.
As the blue triangle is dragged onto the red triangle, the three matching
angles remain the same. The blue and red triangles are obviously not
congruent.
Part 3
Similar triangles
43
Activity – Similar figures
1
The sides double in size but the angles remain the same.
8
6 cm
cm
29°
47°
104°
4 cm
If you double the angles you will get a totally different picture. It is
not even a triangle!
2
a
i
2. The two given sides, 5 mm and 10 mm are matching
sides. 10 mm is double 5 mm. The scale factor that
doubles anything is two. (Triple is 3, quadruple is 4.)
ii
b
i
1
The opposite of enlarging is reducing. The opposite
2
1
scale factor to 2 is .
2
20 mm. The smaller triangle is exactly halfway between
point A and the larger triangle.
ii
40 mm.
You can think two ways. Either add AB and BC together or
double AB because the scale factor is 2.
c
i
40 10
or
5
20
Remember 2 =
ii
44
2
4
6
or or and so on.
1
2
3
5
1 20
,
or
2 40 10
SGS5.2.2 Properties geometrical figures
Activity – Ratios of corresponding sides
1
a
1. When they sit on top of each other they are congruent
triangles. Note that ‘current value of #’ = 1. # is just a
pronumeral like x. In this case # = the scale factor at any
particular moment.
2
b
The triangle vanishes at this point.
c
The ratio of corresponding sides is zero This is because the
sides reduce to nothing at the vanishing point. The scale factor,
(or ‘current value of #)’ = 0, which means it becomes nothing.
d
I
equal
a
i
6
or 6:1
1
ii
distances, equal
iii scale factor
AB 36
If you cancel this fraction down by dividing by 6
=
DE 6
(their highest common factor) you get:
36 6
61
ii
6:1
BC:EF = 54:9. Again, if you cancel this ratio down using
its highest common factor of 9, you get:
6
1
54 : 9 = 6 :1
b
42:7 = 6:1 (By dividing 42 and 7 by 7)
or you can write
c
42 6
= .
7 1
This pair of triangles are similar because all three ratios of their
corresponding sides are the same.
d
1:6, inverse
1
6
is the inverse of
6
1
Part 3
Similar triangles
45
3
a
0.7. Remember there are two ways you can do this:
Method 1
10 × scale factor=7
7
scale factor=
10
Method 2
Transformation
Size of scale factor
Calculation
Reduction:
Scale factor
between 0 and 1
small number
large number
small number
large number
7
=
10
Scale factor=
b
c
2.8
= 2.8 ÷ 4
4
= 0.7
These rectangles are similar because the ratios of their
corresponding sides are equal.
Activity – Writing equations
1
a
EF DE
=
BC AB
side of small shape
EF
, so you must follow the
=
BC corresponding side of large shape
same order. That is:
DE : AB = other side of small shape: corresponding side of
large shape.
b
46
x 12
=
50 20
SGS5.2.2 Properties geometrical figures
c
x = 30 cm.
Multiply both sides by 50.
x 12
50 × = × 50
50 20
∴x =
12
× 50/ You can cross off the zeros because you can
20/
divide both 50 and 20 by its highest common factor, 10.
12
×5
2
= 6 × 5 (12 ÷ 2 = 6 )
= 30 cm
x=
There is another way to solve this problem. You could find the
scale factor first and then apply it to the side that corresponds to
x, like this:
x is on the smaller figure so the scale factor is less than one.
Remember, this means that you should put the small number
12
over the bigger number, like this. Scale factor=
or 12:20.
20
3
These can be reduced to
or 3:5, by dividing by the highest
5
common factor of 4. (That is, 12 ÷ 4 = 3 and 20 ÷ 4 = 5 )
This means that x must be
3
of 50 cm.
5
3
× 50 = 30 cm
5
2
a
Redrawing diagrams may take time but you will make less
mistakes.
A
x
D
10 m
m
A
E
7 mm
m
B
Part 3
Similar triangles
C
12 m
47
b
There are four equations you could write (eight if you reverse
them.) These are the best two because the unknown, x, is in the
top left position of the equation.
7
x
=
10 12
or
x 10
=
7 12
10 12
will give you the same answer
=
7
x
but it takes more time and is more difficult, so avoid using it.
Solving an equation like
c
x = 5.83 or you could just write x ≈ 5.8 mm . As you can see,
not all equations give nice easy numbers.
Both equations give the same answer because 7 ×10 = 10 × 7
7
x
=
10 12
7 ×10
x=
12
x 10
=
7 12
10 × 7
x=
12
Now use a calculator to find x. 10
7
÷
12
=
Activity – How high is that?
1
Results in the table and answers will vary. However your equation
will follow the ratios below the diagram.
B
E
A
D
C
BC AC
which is the same as:
=
ED AD
height of object distance between edge of shadow and base of object
=
distance between first and second point on the ground
your height
You could write either
or you could write
48
BC ED
.
=
AC AD
SGS5.2.2 Properties geometrical figures
Remember that the height of the object is the first thing you should
write. This puts it in the top left position. All other equations are
more difficult to solve.
Part 3
Similar triangles
49
Additional resources – Part 3
Use these guided steps to assist you with the activity on rotations,
translations and reflections in the preliminary quiz question 3d.
First three steps for opening Wingeom.
Wingeom allows you to click and drag points in a figure and make
animations move faster or slower. Wingeom was designed by R Parris at
Exeter University and has been donated for use in this material.
(Note: this software will not operate on Macintosh computers.)
You will need a Windows®-based computer.
Every time you use Wingeom, you will need to follow the first three
steps below.
50
Step 1
Open Wingeom after downloading Wingeom from the link for
this part on the CLI web site. The following window should
appear. (Don’t enlarge this window too much as some
computers do not like it!)
Source
© Wingeom
SGS5.2.2 Properties geometrical figures
Step 2
Go to the menu at the top of Wingeom, click on ‘Window’, drag
down to ‘2-dim’ and release your mouse. This means that you
have selected two-dimensional drawings such as squares
and triangles.
Source
© Wingeom
Step 3
Click on ‘File’, drag down to ‘open’ and release your mouse.
If you have a window like the one below, covering part of the diagram or
text, simply drag it away by the blue border.
Source © Wingeom
Part 3
Similar triangles
51
If some of the text is missing from the window or squashed on top of
itself, you will need to drag the edge of the window to make it bigger.
If you move your mouse to the edge, a vertical or horizontal arrow
appears like this:
Click and drag this arrow to enlarge the window.
52
SGS5.2.2 Properties geometrical figures
Part 3
Similar triangles
53
Exercises – Part 3
Exercises 3.1 to 3.4
Name
___________________________
Teacher
___________________________
Exercise 3.1 – Similar figures
Step1
Open Wingeom, following the first three steps provided for you
in the additional resources then download the files for this
exercise from <http://www.cli.nsw.edu.au/Kto12>. Select
Mathematics then Stage 5.2 and follow the links to resources for
this unit SGS5.2.2 Properties of geometrical figures, Part 3.
Step 2
Highlight the file called ‘Transformations’ and click on open.
Your window should look like this.
Source
54
© Wingeom
SGS5.2.2 Properties geometrical figures
1
a
Name the transformation shown in the screen shot above from
Wingeom.
___________________________________________________
Step 3
Follow the instructions on the screen and answer the
questions below.
b
What transformations move the original shape to the red shape?
___________________________________________________
___________________________________________________
c
How many transformations are there? ____________________
Step 4
You may change the original shape by dragging its corners and
then clicking on ‘autocyc’ again if you wish to explore further.
Step 5
Close Wingeom. (Remember, if you have saved your CD onto
your computer, click on ‘No’ when it asks you to save.
2
a
Use a ruler to measure the following lengths in the diagram
below. Write these measurements in the spaces provided, then
answer the question following each measurement.
F
A
V
C
B
E
i
Measure VE and VB._______________________________
ii
What do you notice about these two lengths? ____________
________________________________________________
iii Measure VC and VF _______________________________
Part 3
Similar triangles
55
iv
Circle the correct statement.
VC
=2
VF
b
VF
=2
VC
v
Measure CB and FE. _______________________________
vi
Complete the equation. FE:___ = 2:___.
Complete the enlargement above by following these steps.
Measure VA
Double this length
Measure this new length along VA and mark a new point D.
Complete the enlarged triangle DEF
3
Use the diagram below to answer the following questions.
(This diagram is not drawn to scale, so you cannot use a ruler.)
G
m
5m
D
E
A
F
25
mm
C
20
mm
B
a
Use the 20 mm and 5 mm lengths to determine the scale factor
used for the:
i
enlargement______________________________________
ii
reduction ________________________________________
b
How long is AB? _____________________________________
c
If BC is 28 mm long, how long is the matching side of the
smaller quadrilateral? __________________________________
d
i
Write down a pair of possible lengths for ED and GF.
________________________________________________
56
SGS5.2.2 Properties geometrical figures
Exercise 3.2 – Scale factors
1
A scale factor of 6
a
1
is used to change the figure below.
2
3 mm
4 mm
5m
m
i
Is this an enlargement or a reduction? __________________
ii
How long are all the sides of the new figure?
________________________________________________
________________________________________________
________________________________________________
b
Write down a scale factor that would reduce the figure above.
___________________________________________________
Calculate the length of the side of this reduced figure that
corresponds to 5 mm.
___________________________________________________
c
The figure below is similar to the figure above. (It is not drawn
to scale.)
17
.5
i
mm
10.5 mm
14 mm
Is this an enlargement or a reduction of the original triangle
at the beginning of the question?______________________
Part 3
Similar triangles
57
ii
Use the the pair of corresponding sides, 4 mm and 14 mm,
to calculate the scale factor that was used to create this new
triangle above.
________________________________________________
________________________________________________
________________________________________________
iii Show that the other two sides had the same scale factor used
on them.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
2
A scale factor of
3
is used on this triangle to create a new triangle.
4
E
4.5
cm
8
D
cm
C
B
5 cm
a
A
Circle the approximate position where you should draw the
new triangle.
A
B
C
D
b
Draw the new triangle in approximately the correct place.
c
i
E
Calculate the length of the new side that corresponds to the
8 cm side of the original triangle.
________________________________________________
________________________________________________
58
SGS5.2.2 Properties geometrical figures
ii
Only one of the triangles below is similar to the triangle
above.
m
6c
m
m
cm
c
2.9
5
3.37
6c
3.75 cm
Trangle 1
3.5 cm
Trangle 2
Write the calculations that you need to do in order to find
out which triangle is similar.
________________________________________________
________________________________________________
iii Complete:
Triangle _____ is similar to the original triangle above
because the scale factor used to reduce each ____________
________________________________________________
3
Answer the following questions using the same diagram drawn at the
beginning of the previous question.
a
If you draw a similar triangle at point D, what scale factor does
it have, compared to the original triangle? __________________
b
Where would you draw a similar triangle with scale factor = 1?
___________________________________________________
c
What would be the scale factor of a triangle drawn at point E?
___________________________________________________
Part 3
Similar triangles
59
Exercise 3.3 – Ratios of corresponding sides
1
The following questions refer to the diagram below.
C
E
A
D
B
a
F
Explain why these triangles are similar.
___________________________________________________
b
What is the corresponding side to DF? ____________________
c
If the ratio, BC:EF = 4:3, what is the ratio,
i
AC:DF? _________________________________________
ii
DF:AC? _________________________________________
iii Calculate
Use the two rectangles below to answer the following questions.
8m
6m
7m
5m
2
AC DF
_______________________________
×
DF AC
a
Are all corresponding angles equal? ______________________
b
Show that the ratios of the corresponding sides are different.
___________________________________________________
___________________________________________________
___________________________________________________
60
SGS5.2.2 Properties geometrical figures
c
Explain why these ratios prove the rectangles are not similar.
___________________________________________________
___________________________________________________
c
Write a pair of dimensions on the rectangle below that make it
similar to the first rectangle above.
3
Explain why any two squares or any two equilateral triangles are
always similar. ___________________________________________
_______________________________________________________
_______________________________________________________
4
Answer the following questions about the pairs of isosceles
triangles below.
6m
m
7m
m
a
5 mm
4 mm
6m
13.
m
5m
m
b
9 mm
i
Part 3
Similar triangles
4 mm
Which pair of isosceles triangles are similar? ____________
61
Show your working below that helped you to come to this
conclusion.
________________________________________________
________________________________________________
________________________________________________
ii
Which pair of isosceles triangles have equal angles? ______
Why? ___________________________________________
iii Explain why all isosceles triangles are not similar.
________________________________________________
________________________________________________
________________________________________________
62
SGS5.2.2 Properties geometrical figures
Exercise 3.4 – Writing equations
1
Answer questions about the following pair of similar triangles.
D
B
x
F
C
A
E
a
Writing x in the top left position, write three correct equations
using equivalent ratios of corresponding sides.
___________________________________________________
___________________________________________________
___________________________________________________
b
Choose one of your equations above to calculate the value of x,
given that AC = 25 m, BC = 7 m and EF = 17.5 m.
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Similar triangles
63
2
In the diagram below BC//DE.
A
12
cm
50°
B
45°
C
8c
m
E
m
28 c
D
a
What is the size of ∠ ADE? _____________________________
b
Explain why ∠ AED = 50 0 _____________________________
___________________________________________________
c
Explain why ∆ ABC is similar to ∆ ADE.__________________
___________________________________________________
d
Draw ∆ ABC and ∆ ADE as separate triangles. Include correct
dimensions.
e
Calculate the length of BC.
___________________________________________________
___________________________________________________
___________________________________________________
64
SGS5.2.2 Properties geometrical figures
3
In which pairs of triangles below can you find the value of x?
(Do not calculate the answers.) ______________________________
Explain why in each case. __________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
a
2.5 m
x
8
m
7m
2 cm
b
7
4c
6c
m
m
x cm
cm
5
cm
7m
c
x°
m
120°
Part 3
Similar triangles
14
m
10.5 m
19
65
Exercise 3.5 – How high is that?
1
a
Copy your results from the practical task onto the table and
diagram below.
Length of pace 1
Length of pace 2
Length of pace 3
Total length of first three paces
Your average pace length
Distance between first and second point on the ground
Distance between edge of shadow and base of object
My height
b
Rewrite your equation and solution here.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
Describe any problems that you may have had that could have
affected your results.
___________________________________________________
___________________________________________________
___________________________________________________
66
SGS5.2.2 Properties geometrical figures
2
John, who is 162 cm tall, wants to measure the height of a flat-roofed
building. He waits for a sunny day, then when he sees a clear
shadow of the building on the ground, he marks where the shadow
ends. John then stands at this point, with his back to the Sun, and
walks backwards until his head is just inside the shadow line of the
building. He marks the point where he is standing. The distance
between these two points is 2.7 m and the distance between the first
point and the building (in the same line as the two points) is 20 m.
a
Write all of his measurement on this diagram.
b
Calculate the height of the building.
___________________________________________________
___________________________________________________
___________________________________________________
c
Ivan, who is 175 cm tall, calculated the height of the same
building. Describe the differences between Ivan’s diagram and
John’s.
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Similar triangles
67
3
Ivan wanted to measure the height of the tree just beyond his house
because he thought it may be the tallest in the town. Ivan had
difficulty taking measurements because there was so much
undergrowth. His diagram is below.
1.75 m
0.75 m
undergrowth
23.7 m
Discuss why these measurements would give an inaccurate height of
the tree.
_______________________________________________________
_______________________________________________________
68
SGS5.2.2 Properties geometrical figures