Ch 02

Pythagoras’
theorem

2
George is a builder and
needs to be sure that the
walls in his building are
square with the floor. This
means that the wall and
floor meet at right angles.
The walls are 240 centimetres
high. The builder runs a string
from the top of the wall to a
point on the floor 1 metre
from the foot of the wall. The
string is measured as being
260 centimetres long. Are the
walls square with the floor?
This chapter looks at
working with right-angled
triangles and, in particular,
using a relationship between
the sides of a right-angled
triangle.

44 Maths Quest 9 for Victoria

Right-angled triangles
This chapter investigates one of the most important ideas in geometry related to right-
angled triangles. Triangles can be classified according to the length of their sides. They
can be classified as either equilateral, isosceles or scalene. Triangles can be classified by
their angles as well. A triangle with one right angle is said to be a right-angled triangle.

Types of triangles

Isosceles Equilateral Scalene Right-angled

In any triangle, the longest side is opposite the largest angle; so Hy
in a right-angled triangle, the longest side is opposite the right po
ten
us
angle. This side has a special name. It is called the hypotenuse. e

The theorem, or rule, that you will learn more about was named after an ancient
Greek mathematician, called Pythagoras (580–501 BC). It will enable you to solve all
kinds of practical problems related to right-angled triangles.
The following exercise will help you to understand what Pythagoras has been
credited with discovering 2500 years ago.

remember
remember
1. The longest side of a right-angled triangle is called the hypotenuse.
2. The hypotenuse is always situated opposite the right angle.

2A Right-angled triangles

2.1 1 For each of the following triangles, carefully measure the length of each side, in milli-
HEET metres, and record your results in the table which follows. Note that the hypotenuse is
SkillS

always marked as c.
a a b c c

a
Pythagoras’ c
b b
theorem b
c

a

Chapter 2 Pythagoras’ theorem 45
d a e f b

a

b a
c c
c

b

g a h a

b
c
b
c

a b c d e f g h

a
b
c
a2
b2

a 2 + b2
c2

2 What do you notice about the results in the table in question 1?
3 On a sheet of paper, carefully draw 6 right-angled triangles of different sizes. You will
need to use a protractor, set square or template to make sure the triangles are right-
angled. Carefully measure the sides of each triangle, and complete a table like the one
in question 1. What do you notice about these results?
4 Now draw some triangles that are not right-angled, measure their sides and complete
the same kind of table. Remember to label the longest side as c. What do you notice
this time?
5 For this activity you will need graph paper, coloured pencils, glue and scissors.
a On a sheet of graph paper, draw a right-angled triangle with a base of 4 cm and a
height of 3 cm.
b Carefully draw a square on each of the three sides of the triangle and mark a grid on
each so that the square on the base is divided into 16 small squares, while the
square on the height is divided into 9 small squares.


c Colour the square on the base, and the square
on the height in different colours as shown at
right so that you can still see the grid lines.
d Carefully cut out the two coloured squares
from the triangle.
e Now stick the larger of the coloured squares
on the uncoloured square of the triangle (the 3 cm
square on the hypotenuse). 4 cm
f Using the grid lines as a guide, see if you can
cut the smaller square up and fit it on the
remaining space. The two coloured squares
should have exactly covered the third square.
g What do you notice about the hypotenuse and
the other two sides of a right-angled triangle?

History of mathematics
P Y T H AG O R A S ( c . 5 8 2 – 5 0 0 BC)

During his life . . . about numbers, but it is impossible to sort out what
Confucius is born. he discovered himself and what was discovered by
The Persians invade others who carried on his work. His ideas influenced
Egypt. the thinking of the Greek philosopher, Plato, who
Greek city-states lived from about 430–350 BC and the followers of
develop and their neoplatonism in the third century AD.
citizens have a Pythagoras is credited with the discovery of what
say in is now known as Pythagoras’ theorem, which states
government. that in a right-angled triangle, the square on the
hypotenuse (long side) is equal to the sum of the
Pythagoras was a squares on the other two sides. This can be
Greek expressed as c2 = a2 + b2.
philosopher and Pythagorean triads (or
mathematician. Little triples) are sets of
is known about him numbers that obey c2
because he lived so long Pythagoras’ theorem.
ago and he left no written work. We do know that Other people knew of a2
c a
he was born on the island of Samos and that as a this idea long before he
b
young man he travelled to Egypt and Babylonia announced it and there is a
(Mesopotamia) where he learned much of his Babylonian tablet known as
‘Plimpton 322’ (believed to b2
mathematics and developed an interest in
investigating it further. When he was 50 years of have been made about 1500
age, he settled in Crotona, a Greek colony in years before Pythagoras was
southern Italy, where he established a religious born) which has a set of the values we now call
community. The members believed in the doctrine Pythagorean triads.
of the transmigration of souls and thought it was It is also thought that Pythagoras discovered that
important to lead a pure and moral life to raise the musical notes have a mathematical pattern.
soul to a higher level for the next life. Questions
Pythagoras and his followers believed that the way 1. Where was Pythagoras born?
to understand the world was through numbers. The 2. What is the formula for his famous theorem?
equilateral triangle seems to have been particularly 3. What is a Pythagorean triad?
significant and was treated as an object of religious 4. Which famous Greek philosopher was
veneration. He probably made several discoveries influenced by Pythagoras?

Using Pythagoras’ theorem
Pythagoras was perhaps the first mathematician to recognise and investigate a very
important property of right-angled triangles. The theorem, or rule, named after him
states that:

In any right angled triangle, the square of the hypotenuse is equal to the sum of
the squares of the other two sides. The rule is written as c2 = a2 + b2 where a and
b are the two shorter sides and c is the hypotenuse.

The hypotenuse is the longest side of a right-angled triangle c
a
and is always the side that is opposite the right angle.
b

Pythagoras’ theorem gives us a way of finding the length of
the third side in a triangle, if we know the lengths of the two x
other sides. 4

Finding the hypotenuse 7
We are able to find the length of the hypotenuse when we are given the length of the
two shorter sides by substituting into the formula c2 = a2 + b2.

WORKED Example 1
For the triangle at right, find the length of the hypotenuse, x,
correct to 1 decimal place.
x
4

THINK WRITE 7

1 Copy the diagram and label the sides a,
b and c. Remember to label the
c=x
hypotenuse as c. a=4

b=7

2 Write Pythagoras’ theorem. c2 = a 2 + b 2
3 Substitute the values of a, b and c into x2 = 4 2 + 7 2
this rule and simplify. = 16 + 49
= 65
4 Calculate x by taking the square root x = 65
of 65. Round your answer correct to x = 8.1
1 decimal place.

In many cases we are able to use Pythagoras’ theorem to solve practical problems. We
can model the problem by drawing a diagram, and use Pythagoras’ theorem to solve the
right-angled triangle. We then use the result to give a worded answer.


WORKED Example 2
A ﬁre is on the twelfth ﬂoor of a building. A child needs to be rescued from a window that
is 20 metres above ground level. If the rescue ladder can be placed no closer than 6 m from
the foot of the building, what is the minimum length ladder needed to make the rescue?
Give your answer correct to the nearest metre.
THINK WRITE
1 Draw a diagram
and label the sides
a, b and c.
a c
Remember to 20 m x
label the
hypotenuse
as c. 6m
b

2 Write Pythagoras’ c2 = a2 + b2
theorem.
3 Substitute the x2 = 202 + 62
values of a, b and c = 400 + 36
into this rule and = 436
simplify.
4 Calculate x by taking x = 436
the square root of 436. = 20.8806
Round your answer ≈ 21
correct to the nearest
metre.
5 Give a worded The ladder
answer. needs to be
21 metres long.

remember
remember
1. The hypotenuse is the longest side of the triangle and is opposite the right
angle.
2. The length of the hypotenuse can be found if we are given the length of the two
shorter sides by using the formula: c2 = a2 + b2.
3. Worded problems can be solved by drawing a diagram and using Pythagoras’
theorem to solve the problem.
4. Worded problems should be given a worded answer.


2B Using Pythagoras’ theorem

WORKED 1 For the following triangles, ﬁnd the length of the hypotenuse, x, correct to 1 decimal 2.2
Example
place. HEET
1

SkillS
a b c
x x 7
x
3 5 L Spread
XCE

sheet
E
24
4 12 Finding the
d e 896 f 1.3 length of the
x hypotenuse
6.2 Math
17.5

cad
742 x x
12.2 Pythagoras’
theorem

2 For each of the following triangles, ﬁnd the length of the hypotenuse, giving answers GC pro
correct to 2 decimal places.

gram
a 4.7 b 19.3 c Pythagoras’
theorem
804
6.3
27.1
562

d e 0.9 f 152

7.4
87
10.3
2.7

3 A right-angled triangle has a base of 4 cm and a height of 12 cm. Find the length of
the hypotenuse to 2 decimal places.
4 Find the lengths of the diagonals of squares that have side lengths of:
a 10 cm b 17 cm c 3.2 cm
5 What is the length of the diagonal of a rectangle whose sides are:
a 10 cm and 8 cm? b 620 cm and 400 cm? c 17 cm and 3 cm?
6 An isosceles triangle has a base of 30 cm and a height of 10 cm. Find the length of the
two equal sides.
7 A right-angled triangle has a height of 17.2 cm, and a base that is half the height. Find
WORKED
the length of the hypotenuse, correct to 2 decimal places.
Example
8 A ladder leans against a vertical wall. The foot of the ladder is 1.2 m from the wall,
2
and the top of the ladder reaches 4.5 m up the wall. How long is the ladder?


9 A flagpole, 12 m high, is supported by three wires, attached from the top of the pole
to the ground. Each wire is pegged into the ground 5 m from the pole. How much
wire is needed to support the pole? 3.8 km

me 10 Sarah goes canoeing in a large lake. She paddles 2.1 km to
E ti
the north, then 3.8 km to the west. Use the triangle at right
GAM

2.1 km
Pythagoras’ to find out how far she must then paddle to get back to her
theorem — starting point in the shortest possible way.
001 Starting point

11 A baseball diamond is a square of side length 27 m. When a runner on first base tries
to steal second base, the catcher has to throw the ball from home base to second base.
How far is that throw?

Second base

27 m

Catcher

Finding a shorter side
Sometimes a question will give you the length of the hypotenuse and ask you to find
one of the shorter sides. In such examples, we need to rearrange Pythagoras’ formula.
Given that c2 = a2 + b2, we can rewrite this as:
a2 = c2 − b2
or b2 = c2 − a2.
It is easier to know what to do in this case if you remember that:

finding a short side means subtract.

WORKED Example 3
Find the length, correct to 1 decimal place, of the unmarked side of
the triangle at right.
14 cm
THINK WRITE 8 cm
1 Copy the diagram and label the sides a, b
and c. Remember to label the hypotenuse a
as c.
c = 14

b=8

2 Write Pythagoras’ theorem for a shorter side. a2 = c2 − b2
3 Substitute the values of a, b and c into this a2 = 142 − 82
rule and simplify. = 196 − 64
= 132
4 Find a by taking the square root of 132. a = 132
Round to 1 decimal place. = 11.5 cm

Practical problems may also involve you being required to ﬁnd the shorter side of a
right-angled triangle.

WORKED Example 4
A ladder that is 4.5 m long leans up against a vertical wall. The foot of the ladder is 1.2 m
from the wall. How far up the wall does the ladder reach? Give your answer correct to
1 decimal place.
THINK WRITE
1 Draw a diagram and label the sides a, b and
c. Remember to label the hypotenuse as c. c = 4.5 m
a

b = 1.2 m
2 Write Pythagoras’ theorem for a shorter side. a2 = c2 − b2
3 Substitute the values of a, b and c into this a2 = 4.52 − 1.22
rule and simplify. = 20.25 − 1.44
= 18.81
4 Find a by taking the square root of 18.81. a = 18.81
Round to 1 decimal place. = 4.3 m
5 Give a written answer. The ladder will reach a height of 4.3 m up
the wall.

Some questions will require you to decide which method is needed to solve the
problem. A diagram will help you decide whether you are ﬁnding the hypotenuse or
one of the shorter sides.


remember
remember
1. A shorter side of the triangle can be found if we are given the length of the
hypotenuse and the other shorter side.
2. When ﬁnding a shorter side, the formula used becomes a2 = c2 − b2 or
b2 = c2 − a2.
3. On your diagram check whether you are ﬁnding the length of the hypotenuse or
one of the shorter sides.

2C Finding a shorter side

reads
L Sp he
WORKED 1 Find the length, correct to 1 decimal place, of the unmarked side in each of the
Example
et
EXCE

3
following triangles.
Finding a b c
the length of 14 3.2
the shorter side 10
d
hca
17 8.4
Mat

8
Pythagoras’
theorem
d e 1.2 f 51.8
ogram 382
GC pr

Pythagoras’ 457
theorem 75.2
3.8

2 Find the value of the pronumeral, correct to 2 decimal places.
a a b c
c
1.98 8.4
30.1
47.2
2.56 17.52

b

d 0.28 e f
2870

d 468
1920 f
e
0.67

114
For questions 3 to 13, give your answers correct to 2 decimal places.

3 The diagonal of the rectangular sign at right is
34 cm. If the height of this sign is 25 cm, find
the width.
4 The diagonal of a rectangle is 120 cm. One side
has a length of 70 cm. Find:
a the length of the other side
b the perimeter of the rectangle
c the area of the rectangle.
5 An equilateral triangle has sides of length 20 cm. Find
the height of the triangle.
6 The road sign shown at left is in the form of an
equilateral triangle. Find the height of the sign and,
hence, find its area.
76 cm
WORKED 7 A ladder that is 7 metres long leans up against a vertical
Example
4
wall. The top of the ladder reaches 6.5 m up the wall.
How far from the wall is the foot of the ladder?

8 A tent pole that is 1.5 m high is to be supported
by ropes attached to the top. Each rope is 2 m
long. How far from the base of the pole can each
rope be pegged?

9 A kite is attached to a string 150 m long. Sam holds the
150 m
end of the string 1 m above the ground, and the
horizontal distance of the kite from Sam is 80 m as
80 m shown at left. How far above the ground is the kite?
1m 10 Ben’s dog ‘Macca’ has wandered onto a frozen pond,
and is too frightened to walk back. Ben estimates that
the dog is 3.5 m from the edge of the pond. He finds a
plank, 4 m long, and thinks he can use it to rescue
Macca. The pond is surrounded by a bank that is 1 m
high. Ben uses the plank to make a ramp for Macca to
walk up. Will he be able to rescue his dog?
11 Penny, the carpenter, is building a roof for a new house. The roof
has a gable end in the form of an isosceles triangle, with a base 7.5 m 7.5 m
of 6 m and sloping sides of 7.5 m. She decides to put 5 evenly
spaced vertical strips of wood as decoration on the gable as shown
at right. How many metres of this decorative wood does she need?
6m
12 Wally is installing a watering system in his garden. The
pipe is to go all around the edge of the rectangular
56 cm garden, and have a branch diagonally across the garden.
The garden measures 5 m by 7.2 m. If the pipe costs
40 cm $2.40 per metre (or part thereof), what will be the total
cost of the pipe?
13 The size of a rectangular television screen is given by
the length of its diagonal. What is the size of the screen
at left?


Shortest path
A surf lifesaving contest involves swimming A
from marker A to the shoreline BC, then on to D
100 m 80 m
marker D as shown in the diagram at right.
Suppose a lifesaver swims the course A–E–D B E C
where E is 60 m from B. 60 m
150 m

1 What total distance will the lifesaver swim?
2 Suppose point E was 100 m from B. Determine how far the lifesaver would
swim in total.
3 Suppose point E is halfway between B and C. How far would the lifesaver swim
in total?
4 Complete the table below.
Length BE Length AE Length ED Total length
(m) (m) (m) (AE + ED) (m)
50
60
70
80
90
100
110
5 Use the data in the table to plot a graph of
(AE + ED) (m)

length BE (horizontal axis) versus total
Total length

240
length (AE + ED) (vertical axis). 238
6 What is the shortest distance the lifesaver 236
could swim from marker A to the shoreline 234
then to marker D?
7 Draw a scale diagram showing the path taken 50 70 90 110
Length BE (m)
by the lifesaver to swim this shortest distance.
Measure the angles AEB and DEC. What do you notice?

The shortest distance covered by the lifesaver
obeys the Law of Reflection. The angle of A
D
incidence (AEB) equals the angle of reflection 100 m
80 m
(DEC). We can use this law and a scale
diagram to find the point, E, for situations B E C
where the markers are at different distances D'
from the shoreline. Reflect the line segment,
DC, about the line, BC (shown dotted), then
connect A to D′. The intersection point with the shoreline gives the position of E.
8 Use the Law of Reflection to solve A
the following problem. Remember G
to use a scale diagram. D
This time the race involves
100 m 80 m
touching the shoreline in two places
E and F. What is the shortest 60 m
distance the lifesaver should swim? B E F C
100 m
200 m

1
1 True or false?
a The hypotenuse is the longest side of a right-angled triangle. x
b The hypotenuse forms part of the right angle. 2 cm
c Pythagoras’ theorem involves the formula a2 − b2 = c2.
2 Find the length of the hypotenuse of the triangle above right, 5 cm
correct to 2 decimal places. 11 m
3 Find the value of the pronumeral in the triangle at right,
correct to 1 decimal place. x
16 m
4 multiple choice
The length of a diagonal of a square which has sides of 8 mm is:
A 10.5 mm B 11.31 mm C 17.3 mm D 19.6 mm E 22.23 mm
5 A roof is 3 m above the ground. A ladder is placed 1 m from the foot of the wall. How
long does the ladder need to be in order to reach the roof? (Give your answer correct
to the nearest cm.)
6 A flagpole is supported by a wire that is 5 m long. The wire is pegged 2.5 m from the
foot of the pole. Find the height of the pole, correct to 2 decimal places.
7 The diagonal of a rectangle is 9.2 cm. The length is 6.3 cm. Find the width, correct to
1 decimal place.
8 Find the area of the rectangle in question 7.
9 Keith drives 5.3 km east, then 9.2 km north. How far in, metres, is he from his starting
point? (Give your answer correct to 1 decimal place.)
10 An isosceles triangle has sloping edges equal to 10 cm and a base equal to 5 cm.
Calculate the height of the triangle, correct to 1 decimal place.


Working with different units
When we use Pythagoras’ theorem, we are usually working with a practical situation
where measurements have been given. In any calculation, it is essential that all of the
measurements are in the same units (for example, cm).
Do you remember the relationship between the units of length?

10 mm = 1 cm
100 cm = 1 m To change to a larger unit, divide by the conversion factor.
To change to a smaller unit, multiply by the conversion factor.
1000 m = 1 km

For example, to change kilometres to metres, multiply by 1000.
To change centimetres to metres, divide by 100.
The following chart shows all the conversions of length that you are likely to need.

÷ 10 ÷ 100 ÷ 1000

millimetres centimetres metres kilometres
(mm) (cm) (m) (km)

× 10 × 100 × 1000

When using Pythagoras’ theorem, always check the units given for each measurement.
If necessary, convert all measurements to the same units before using the rule.

WORKED Example 5
Find the length, in mm, of the hypotenuse of a right-angled triangle if the 2 shorter sides
are 7 cm and 12 cm.
THINK WRITE
1 Draw a diagram and label the sides a, b
and c. Remember to label the c
hypotenuse as c. a = 12 cm

b = 7 cm
2 Check that all measurements are in the
same units. They are.
3 Write Pythagoras’ theorem for the c2 = a2 + b2
hypotenuse.
4 Substitute the values of a and b into this c2 = 122 + 72
rule and simplify. = 144 + 49
= 193
5 Find c by taking the square root. Give c = 193
the units in the answer. = 13.89 cm
6 Check the units required in the answer = 13.89 × 10 mm
and convert if necessary. = 138.9 mm

WORKED Example 6
The hypotenuse and one other side of a right-angled triangle are 450 cm and 3.4 m
respectively. Find the length of the third side, in cm, correct to the nearest whole number.
THINK WRITE
1 Draw a diagram and label the sides
a, b and c. Remember to label the c = 450 cm
hypotenuse as c. a

b = 3.4 m
2 Check that all measurements are in 3.4 m = 3.4 × 100 cm
the same units. They are different, = 340 cm
so convert 3.4 m into cm.
3 Write Pythagoras’ theorem for a a2 = c2 − b2
shorter side.
4 Substitute the values of b and c a2 = 4502 − 3402
into this rule and simplify. = 202 500 − 115 600
= 86 900
5 Find a by taking the square root of a = 86 900
86 900. Round to the nearest whole = 295
number.
6 Give a worded answer. The third side will be approximately 295 cm long.

remember
remember
1. When using Pythagoras’ theorem, always check the units given for each
measurement.
2. If necessary, convert all measurements to the same units before using the rule.

Working with different
2D units
Where appropriate, give answers correct to 2 decimal places.
WORKED 1 Find the length, in mm, of the hypotenuse of a right-angled triangle, if the two shorter 2.3
Example HEET
sides are 5 cm and 12 cm.
SkillS

5
2 Find the length of the hypotenuse of the following right-angled triangles, giving the
answer in the units speciﬁed.
a Sides 456 mm and 320 mm, hypotenuse in cm. XCE
L Spread
b Sides 12.4 mm and 2.7 cm, hypotenuse in mm.
sheet
E

c Sides 32 m and 4750 cm, hypotenuse in m. Pythagoras’
d Sides 2590 mm and 1.7 m, hypotenuse in mm. theorem
e Sides 604 cm and 249 cm, hypotenuse in m.
f Sides 4.06 km and 4060 m, hypotenuse in km.
g Sides 8364 mm and 577 cm, hypotenuse in m.
h Sides 1.5 km and 2780 m, hypotenuse in km.


WORKED 3 The hypotenuse and one other side of a right-angled triangle are given for each case
Example
below. Find the length of the third side in the units specified.
6
a Sides 46 cm and 25 cm, third side in mm.
b Sides 843 mm and 1047 mm, third side in cm.
c Sides 4500 m and 3850 m, third side in km.
d Sides 20.3 cm and 123 mm, third side in cm.
e Sides 6420 mm and 8.4 m, third side in cm.
f Sides 0.358 km and 2640 m, third side in m.
g Sides 491 mm and 10.8 cm, third side in mm.
h Sides 379 000 m and 82 700 m, third side in km.
4 Two sides of a right-angled triangle are given. Find the third
side in the units specified. The diagram shows how each triangle c
is to be labelled. Remember: c is always the hypotenuse. a
a a = 37 cm, c = 180 cm, find b in cm.
b a = 856 mm, b = 1200 mm, find c in cm. b
c b = 4950 m, c = 5.6 km, find a in km.
d a = 125 600 mm, c = 450 m, find b in m.
e a = 0.0641 km, b = 0.153 km, find c in m.
f a = 639 700 cm, b = 2.34 km, find c in m.
4 cm
5 multiple choice
a What is the length of the hypotenuse in this triangle? 3 cm
A 25 cm B 50 cm C 50 mm
D 500 mm E 2500 mm
82 cm
b What is the length of the third side in this triangle?
A 48.75 cm B 0.698 m C 0.926 m 43 cm
D 92.6 cm E 69.8 mm

c The most accurate measure for the length of the third
side in the triangle at right is:
A 4.83 m B 23.3 cm C 3.94 m 5.6 m
D 2330 mm E 4826 mm
2840 mm

d What is the length of the third side in this triangle?
A 34.71 m B 2.97 m C 5.89 m
D 1722 cm E 4.4 m 394 cm

4380 mm
6 A rectangle measures 35 mm by 4.2 cm. Find the length of the diagonal in mm.
7 A sheet of A4 paper measures 210 mm by 297 mm. Find the length of the diagonal in
centimetres.
8 A rectangular envelope has a length of 21 cm and a diagonal measuring 35 cm. Find:
a the width of the envelope
b the area of the envelope.
9 A right-angled triangle has a hypotenuse of 47.3 cm and one other side of 30.8 cm.
Find the area of the triangle.

10 A horse race is 1200 m. The track is straight, and 35 m wide. How much further than
1200 m will a horse run if it starts on the outside and ﬁnishes on the inside as shown?

Finishing
Starting gate post

1200 m
35 m

11 A ramp is 9 metres long, and rises to a height of 250 cm. What is the horizontal
distance, in metres, between the bottom and the top of the ramp?

12 Sarah is making a gate, which has to be 1200 mm wide. It must be braced with a
diagonal strut made of a different type of timber. She has only 2 m of this kind of
timber available. What is the maximum height of the gate that she can make?
13 A rectangular park is 260 m by 480 m. Danny usually trains by running around the
edge of the park. After heavy rain, two adjacent sides are too muddy to run along,
so he runs a triangular path along the other two sides and the diagonal. Danny does
5 circuits of this path for training. How far does he run? Give your answer in km.
14 A swimming pool is 50 m by 25 m. Peter is bored by his usual training routine, and
decides to swim the diagonal of the pool. How many diagonals must he swim to com-
plete his normal distance of 1200 m?
15 A hiker walks 4.5 km west, then 3.8 km south. How far in metres is she from her SHE
ET 2.1
Work

starting point?
16 A square has a diagonal of 10 cm. What is the length of each side?


What is Bagheera in ‘The Jungle Book’?
Bagheera Jungle Book’
Calculate the lengths of the lettered
sides in the triangles. Place the lengths with their
letters above them, in ascending order in the boxes below.
The letters will spell out the puzzle answer.

Give lengths accurate to 2 decimal places.
130 cm
6m

1.2 m
R 1m
B
9m
L 1.5 m

6m 7m

A
T
6.31 m E
520 cm
155 cm
180 cm
7.2 m N

170 cm
A
2m 6.2 m
4m
13 cm
C
12.98 cm
5.8 m

A
4m K
3m
H
3m 578 cm

P 621 cm
567 cm

Composite shapes
In all of the exercises so far, we have been working with only one right-angled triangle.
Many situations involve more complex diagrams, where the right-angled triangle is not
as obvious. A neat diagram is essential for these questions.

WORKED Example 7 8 cm
Find the length of the side, x. Give your x
4 cm
answer correct to 2 decimal places.
THINK 10 cm WRITE
1 Copy the diagram. On the diagram, create a right-angled triangle 8
and use the given measurements to work out the lengths of 2 sides.
2 Label the sides of your right-angled triangle as a, b and c. c a 4
Remember to label the hypotenuse as c. b
2 8
3 Check that all measurements are in the same units. They are the same.
4 Write Pythagoras’ theorem for the hypotenuse. c 2 = a2 + b2

5 Substitute the values of a, b and c into this rule and simplify. x2 = 4 2 + 2 2
= 16 + 4
= 20
6 Find x by taking the square root of 20. Round your answer correct x = 20
to 2 decimal places. = 4.47 cm
Some situations will involve shapes that contain more than one triangle. In this case it
is a good idea to split the diagram into separate right-angled triangles first.

WORKED Example 8
For the diagram at right, find 15
the length of the sides marked 6 x
x and y to 2 decimal places.
THINK 2 WRITE
y
1 Copy the diagram.
15 a
2 Find and draw any right-angled triangles c a x 6 x
contained in the diagram. Label their sides. c
3 To find an unknown side in a right-angled b b
y 2
triangle, we need to know 2 sides, so find x first.
4 For the triangle containing x, write down a2 = c2 − b2
Pythagoras’ theorem for a shorter side. x2 = 62 − 22
Use it to find x. = 36 − 4
= 32
x = 32
= 5.66
5 We now know 2 sides for the other triangle
because we can substitute x = 5.66.
6 For the triangle containing y, write down b2 = c2 − a2
Pythagoras’ theorem for a shorter side and y2 = 152 − 5.662
use it to find y. = 225 − 32
= 193
y = 193
= 13.89


remember
remember
1. To examine more complex situations involving Pythagoras’ theorem, a diagram
is essential.
2. When a diagram is given, try to create or separate any right-angled triangles
using further diagrams.

2E Composite shapes
8 cm
WORKED
Example
1 Find the length of the side x in the figure at right. x
reads 7 cm
L Sp he
7
et
EXCE

2 For the following diagrams, find the length of the sides
Pythagoras’ 12 cm
theorem marked x.
a 12 b x c x
d x
hca 7 37 20
120
Mat

200
Pythagoras’ 15
52
theorem 100

WORKED 3 For each of the following diagrams, find the length of the sides marked x and y.
d Example
hca a b
8
Mat

Pythagoras’ 10
x 5
theorem DIY 12
8 x
3
y
y 4
c 5 d
y 18

x
x 5
12
20
10 2
7

4 multiple choice
a The length of the diagonal of the rectangle at right is: 12
A 9.7 B 19 C 13.9 D 12.2 E 5

b The area of the rectangle at right is:
A 12.1 B 84.9 C 15.7 D 109.6 E 38 14
7

20
c The value of x in this shape is: x
A 24 B 24.7 C 26 D 38.4 E 10 24

30

d What is the value of x in this figure?
A 5.4 B 7.5 C 10.1 D 10.3 E 4 x
5

2 7
5 A hobby knife has a blade in the shape of a right-angled trapezium with the sloping
edge 20 mm, and parallel sides of 32 mm and 48 mm. Find the width of the blade and,
hence, the area.
6 Two buildings, 10 and 18 m high, are directly opposite each other on either side of a
6 m wide European street. What is the distance between the top of the two buildings?

18 m
10 m

6m

7 Jess paddles a canoe 1700 m to the west, then 450 m south, and then 900 m to the
east. She then stops for a rest. How far is she from her starting point?
8 A yacht race starts and finishes at A and consists of 6 legs;
AB, BC, CA, AE, EC, CA, in that order as shown in the B 4 km A
figure at right. If AB = 4 km, BC = 3 km and CE = 3 km,
find: 3 km
a AE
b AC
C
c the total length of the race.
3 km
E


9 A painter uses a trestle to stand on in order to paint a ceiling. It consists of 2 stepladders
connected by a 4 m long plank. The inner feet of the 2 stepladders are 3 m apart, and
each ladder has sloping sides of 2.5 m. How high off the ground is the plank?
10 A feature wall in a garden is in the shape of a 4.7 m
trapezium, with parallel sides of 6.5 m and 4.7 m. The
wall is 3.2 m high. It is to have fairy lights around the 3.2 m
perimeter (except for the base). How many metres of
6.5 m
lighting are required?
10 m
11 A garden bed is in the shape of a rectangle, with a triangular
pond at one end as shown in the ﬁgure at right. 5m
Garden
Tony needs to cover the garden to a depth of 20 cm with Pond
topsoil. How much soil (in cubic metres) does he need? 3m

12 Katie goes on a hike, and walks 2.5 km north, then 3.1 km east. She then walks
1 km north and 2 km west. How far is she, in a straight line, from her starting point?
13 A rectangular gate is 3.2 m long and 1.6 m high, and 3.2 m
consists of three horizontal beams, and ﬁve vertical
beams as shown in the diagram at right. Each section is
1.6 m
braced with diagonals. How much timber is needed for
the gate?
14 The diagram at right shows the cross-section through a roof. 5200 mm
a Find the height of the roof, h, to the nearest millimetre. h
b The longer supports are 5200 mm long. Find the length
of the shorter supports, to the nearest millimetre. 9000 mm

A 10

15 Find the distance, AB, in the plan of the paddock at right. 10

B
17
1
16 Calculate the value of a, b, c, d. Leave your answers in surd
1
(square root) form. Can you see a pattern? 1
a
b
c 1
d
1

QUEST
S
M AT H

GE

1 If your watch is running 2 minutes late and
EN

loses 5 seconds each hour, how many
CH L hours will it take for your watch to
AL be running 5 minutes late?
2 Sixteen matches are used to create the
pattern shown at right. Remove 4 matches
to leave exactly 4 triangles. Draw your answer.


Career profile
ROB BENSON — Sheet metal w orker
and found the work appealed to me. I make
food-processing machinery for food
manufacturers (like Cadbury). Projects can
take from a week to a month to complete. A
typical day is spent completing part of a
larger project. Occasionally I go on-site to
install machinery.
I need to use mathematics in my job every
day for many tasks; such as interpreting
draftsmen’s plans and calculating costs
associated with each job. In particular, I use
measurements to check that angles are
‘square’. A key formula that helps me with
Name: Rob Benson this is Pythagoras’ theorem.
Profession: Sheet metal worker
Questions
Qualiﬁcations: First Class Sheet Metal 1. What type of machinery does Rob make at
apprenticeship Tripax Engineering?
Employer: Tripax Engineering 2. Explain how Rob can check that angles are
‘square’.
I was offered an apprenticeship in the sheet 3. Find out how you can obtain information
metal working trade soon after I left school about apprenticeships that are available.

Pythagorean triads
Pythagorean triads, or triads for short, are 3 whole numbers that satisfy Pythagoras’
theorem. This means that the 3 numbers could be the sides of a right-angled triangle.
We test Pythagorean triads by ﬁrst calculating c2, then calculating a2 + b2. If both
calculations yield the same result, the original 3 numbers form a Pythagorean triad.

WORKED Example 9
Do the numbers 5, 7 and 10 form a Pythagorean triad?
THINK WRITE
1 Arrange the values so that the largest Let a = 5, b = 7, and c = 10.
will be c, the hypotenuse.
2 Write down Pythagoras’ theorem. c2 = a2 + b2
3 Substitute the value of c in the left-hand c2 = 102
side of the equation and evaluate. = 100
4 Substitute the values of a and b in the a 2 + b2 = 5 2 + 72
right-hand side of the equation and = 25 + 49
evaluate. = 74
5 If both sides of the equation are equal, c2 ≠ a2 + b2
the numbers form a Pythagorean triad. The numbers do not form a Pythagorean triad.


The numbers 3, 4 and 5 form a Pythagorean triad because 32 + 42 = 25 and 52 = 25. If
we multiply each of these 3 numbers by another number, the resulting numbers will
also form a Pythagorean triad. For example, if we multiply 3, 4 and 5 by 5, we get 15,
20 and 25. These 3 numbers satisfy Pythagoras’ theorem.
Check: c2 = 252
= 625
a2 + b2 = 152 + 202
= 225 + 400
= 625
so c = a2 + b2.
2

Algebra can be used to find sets of numbers that are Pythagorean triads. There are
2 ways in which this can be done.
Method 1
Start with any odd number, and make this the shortest length of the triangle.
S2 – 1
Use the formula M = ------------- where S = shortest side and M = middle side, to
-
2
calculate the middle side of the triangle.
You now have two sides of the triangle and can use c2 = a2 + b2 to calculate the third
side (hypotenuse, c).

WORKED Example 10
If the smallest number of a Pythagorean triad is 7, find the middle number and, hence,
find the third number.
THINK WRITE
1 Write down the given information, and S=7
the formula to find the middle side, M. S2 – 1
M = -------------
-
2
Substitute S = 7 into the formula and 72 – 1
2 = -------------
-
evaluate. 2
49 – 1
= --------------
-
2
48
= -----
-
2
= 24
The middle number is 24.
3 Use Pythagoras’ theorem to find the c2 = a2 + b2
third number. = 72 + 242
= 49 + 576
= 625
c = 625
= 25
The third number is 25.
4 State the solution. The Pythagorean triad required is 7, 24, 25.

Why do you think this rule works only when the smallest side is an odd number?

Method 2
This method takes any 2 numbers, and applies a set of rules to them, in order to
generate Pythagorean triads. Select 2 numbers, x and y, with the following rules:
Rule 1 The number chosen for x must be larger than y; that is x > y.
Rule 2 One number must be odd and the other even.
Rule 3 The numbers chosen for x and y must have no common factors. For example,
we could not choose 6 and 9 since they have a common factor of 3.
The Pythagorean triad is given by: 2xy, x2 − y2, x2 + y2.

WORKED Example 11
Find a Pythagorean triad using the values x = 7 and y = 2.
THINK WRITE
1 Write down the values given in the x = 7, y = 2
question.
2 Substitute these values into the 2xy = 2 × 7 × 2
expression for each term. = 28
x2 − y2 = 72 − 22
= 49 − 4
= 45
x2 + y2 = 72 + 22
= 49 + 4
= 53
3 Check that the numbers obtained satisfy c = 532
2

Pythagoras’ theorem. = 2809
Does c2 = a2 + b2? a + b = 282 + 452
2 2

= 784 + 2025
= 2809
Therefore, c2 = a2 + b2.
4 State the solution. The Pythagorean triad required is 28, 45, 53.

remember
remember
1. Pythagorean triads consist of three whole numbers that satisfy Pythagoras’
theorem.
S2 – 1
2. Starting with an odd number, S, we can use the formula M = ------------- to ﬁnd
-
2
the middle number, M, of a Pythagorean triad. The third number can be found
using Pythagoras’ theorem (c2 = a2 + b2).
3. We can produce a Pythagorean triad using chosen values of x and y by
substituting these into the expressions 2xy, x2 – y2, x2 + y2.


2F Pythagorean triads
reads
L Sp he WORKED 1 Do the numbers given below form Pythagorean triads?
Example
et
EXCE

9 a 6, 8, 10 b 5, 12, 13 c 4, 5, 6
Right d 24, 7, 25 e 16, 20, 12 f 14, 16, 30
angle
tester 2 Use the triads below to create three other triads, and check that they satisfy Pythagoras’
d theorem.
hca
Mat

a 3, 4, 5 b 5, 12, 13
Pythagorean
triads WORKED 3 The smallest numbers of four Pythagorean triads are given below. Find the middle
Example
10
number and, hence, find the third number.
a 9 b 11 c 13 d 29
4 What do you notice about the triads in question 3 above?
WORKED 5 For each of the following, find a Pythagorean triad using the given values of x and y. (In
Example
11
each case check that the 3 numbers found satisfy Pythagoras’ theorem.)
a x = 6 and y = 1 b x = 7 and y = 4 c x = 8 and y = 3
d x = 11 and y = 6 e x = 14 and y = 9 f x = 15 and y = 2

ET 2.2 6 Challenge!
SHE
a Does the method used in question 5 work if both x and y are even?
Work

b Does it work if both x and y are odd?
Try some examples of your own.

Will the house stand up?
At the beginning of the chapter, we saw how
George checked if the walls of the house he was
building were ‘square’ with the floor. The wall
was 240 cm high and a string was pegged to the
floor 1 m from the foot of the wall. The string
was then measured as being 260 cm.
1 Draw a diagram of this problem.
2 Convert all measurements in the problem to
centimetres.
3 Do the numbers in part 2 form a Pythagorean
triad?
4 Are the walls square with the floor of the house?
5 George checks another wall in a room with a
height of 3320 mm. Again a string is pegged to
the floor 1 m from the foot of the wall. If the
wall is square with the floor, what
measurement would you expect for the length
of the string?
6 Give 3 more examples of situations where
Pythagoras’ theorem would be useful.


2
3 cm
1 Find the value of the hypotenuse, correct to 1 decimal place.
7 cm

17 m
2 Find the value of the pronumeral, correct to 1 decimal place. x

3 multiple choice 11 m
A road sign is in the form of an equilateral triangle. Each side measures 54 cm. The
height of the triangle is:
A 31.4 cm B 46.8 cm C 55.3 cm D 61.2 cm E 72.3 cm
4 A rectangular garden bed is 190 cm long and the diagonal is 3 m. Find the width of
the garden, in metres, correct to 2 decimal places.
5 Find the area of the garden bed in question 4.
6 True or false?
In a right-angled triangle if the hypotenuse is 890 cm and one other side is 0.5 m, then
the third side is 8.91 m.
7 Do the numbers 5, 12, 13 form a Pythagorean triad?
8 Are the numbers 9, 17, 19 a Pythagorean triad?
9 If the smallest number of a Pythagorean triad is 7, ﬁnd the middle number and, hence,
ﬁnd the third number.
10 Use x = 8 and y = 5 to form a Pythagorean triad.

Pythagoras in 3-D
Many real-life situations involve 3-dimensional (3-D) shapes: shapes with length, width
and height. Some common 3-D shapes used in this section include boxes, pyramids and
right-angled wedges.

Box Pyramid Right-angled wedge

The important thing about 3-D shapes is that in a diagram, right angles may not look
like right angles, so it is important to redraw sections of the diagram in 2 dimensions,
where the right angles can be seen accurately.


WORKED Example 12
Find the length AG in this box. A B

6
C
D
E F
5

H 10 G

THINK WRITE
1 Draw the diagram in 3-D. A B

6
C
D
E F
5

H 10 G
2 Draw, in 2-D, a right-angled triangle A
that contains AG and label the sides.
Only 1 side is known, so we need to
ﬁnd another right-angled triangle to 6
use.
E G
3 Draw EFGH in 2-D and show the E F
diagonal EG. Label the side EG as x.
5 x 5

H 10 G
4 Use Pythagoras’ theorem to ﬁnd EG. c =a +b
2 2 2

x2 = 52 + 102
= 25 + 100
= 125
x = 125
= 11.18
5 Place this information on triangle AEG. A
Label the side AG as y.
y
6

E 11.18 G

6 Solve this triangle for AG. c =a +b
2 2 2

y2 = 62 + 11.182
= 36 + 125
= 161
y = 161
= 12.69
7 Write your answer. The length of AG is 12.69.

WORKED Example 13
A piece of cheese in the shape of a right-angled wedge sits on a table. It has a rectangular
base measuring 14 cm by 8 cm, and is 4 cm high at the thickest point. An ant crawls
diagonally across the sloping face. How far, to the nearest millimetre, does the
ant walk?
THINK WRITE

1 Draw a diagram in 3-D and label the B C
vertices. E 4 cm
Mark BD, the path taken by the ant, with F
A 8 cm
a dotted line. 14 cm D
x
2 Draw in 2-D a right-angled triangle that B
contains BD, and label the sides. Only
one side is known, so we need to ﬁnd
4
another right-angled triangle to use.
D
E
3 Draw EFDA in 2-D, and show the E F
diagonal ED. Label the side ED as x.
8 x 8

A 14 D

4 Use Pythagoras’ theorem to ﬁnd ED. c =a +b
2 2 2

x2 = 82 + 142
= 64 + 196
= 260
x = 260
= 16.12
5 Place this information on triangle BED. B
Label the side BD as y.
y
4

D
E 16.12
6 Solve this triangle for BD. c2 = a2 + b2
y2 = 42 + 16.122
= 16 + 260
= 276
y = 276
= 16.61 cm
7 Check the answer’s units: we need to = 166.1 mm
convert cm to mm; so multiply by 10.
8 Give the answer in worded form, The ant walks 166 mm.
rounded to the nearest mm.


remember
remember
1. Pythagoras’ theorem can be used to solve problems in 3 dimensions (3-D).
2. Some common 3-D shapes include boxes, pyramids and right-angled wedges.
3. To solve problems in 3-D it is helpful to draw sections of the original shape in
2 dimensions (2-D).

2G Pythagoras in 3-D

2.4 WORKED 1 Find the length, AG.
HEET Example
a b c
SkillS

12 A B A B A B

C 10 D C C
reads D D
L Sp he 10.4
et
EXCE

Pythagoras’ E 10 E
F F
theorem
10 7.3

hca
d H 10 G H 8.2 G
E F
Mat

Pythagoras’ 5
theorem H G
5 A B
E 4
2 Find CB in the wedge at right and, hence, find AC. F
D 10 C 7

A B
3 If DC = 3.2 m, AC = 5.8 m, and CF = 4.5 m in the figure
E
at right, find AD and BF. F
D C

V
8
A
4 Find BD and, hence, the height of the pyramid at right. B
8
D
8 C

E
EM = 20 cm

5 The pyramid ABCDE has a square base. The pyramid is
20 cm high. Each sloping edge measures 30 cm. Find the A
B
length of the sides of the base. M
D C


6 The sloping side of a cone is 10 cm and the height is 8 cm.
Find the radius of the base.
8 cm 10 cm

7 An ice-cream cone has r
a diameter across
the top of 6 cm,
and sloping side
of 13 cm. How
deep is the cone?

WORKED 8 A piece of cheese in the shape of a right-angled wedge B C
Example
13
sits on a table. It has a base measuring 20 mm by 10 mm, 4 mm
E
and is 4 mm high at the thickest point, as shown in the F
figure. A fly crawls diagonally across the sloping face. A 20 mm D 10 mm
How far, to the nearest millimetre, does the fly walk?
30
cm

9 Jodie travels to Bolivia, taking with her a
suitcase as shown in the photo. She buys a
carved walking stick 1.2 m long. Will she
be able to fit it in her suitcase for the flight
65 cm
home?

90 cm

10 A desk tidy is shaped like a cylinder, height 18 cm and diameter
10 cm as shown. A pencil that is 24 cm long rests inside. What
length of the pencil is above the top of the cylinder? 18 cm

10 cm

11 A 10 m high flagpole is in the corner of a rectangular
park that measures 240 m by 150 m. 10 m
240 m A
a Find:
iii the length of the diagonal of the park 150 m
iii the distance from A to the top of the pole B
iii the distance from B to the top of the pole.
b A bird flies from the top of the pole to the centre of the park. How far does it fly?


12 A candlestick is in the shape of two cones, joined at the vertices as
shown. The smaller cone has a diameter and sloping side of 7 cm,
and the larger one has a diameter and sloping side of 10 cm. How tall
is the candlestick?

13 The total height of the shape at right is 15 cm. Find the length 15 cm
of the sloping side of the pyramid.
6 cm

14 cm
14 cm

14 A sandcastle is in the
shape of a truncated
cone as shown.
Find the diameter
of the base.
20 cm

30 cm 32 cm

me
E ti
15 A tent is in the shape of a triangular prism, with a height of
GAM

120 cm
Pythagoras’ 120 cm as shown at right. The width across the base of the
theorem — door is 1 m, and the tent is 2.3 m long.
002 Find the length of each sloping side, in metres. Then 2.3 m
ﬁnd the area of fabric used in the construction of the 1m
SHE
ET 2.3 sloping rectangles which form the sides.
Work

16 A cube has sides of x cm. Find, in terms of x, the length of the diagonal from a top
corner to the opposite bottom corner.

Electrical cable
When electricians need to place electrical cable in a room built on a concrete slab,
they need to connect it along the walls or the ceiling. To keep the costs down, they
would try to use the least amount of cabling.

For each of the two rooms shown below, ﬁnd the shortest length of electrical
cable that could be used.
A
1 The electrical cable needs to run from one top corner 3m
(point A) of the room to another corner (point B).
(Hint: First draw the nets of the two rectangles that
contain points A and B. There are two possibilities.) 4m

6m B
2 In this second room the electrical cable needs to 10 m
run from point A (which is situated in the middle A
of the wall where it meets the ceiling) to point B.

8m

5m
B

Ch 02

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