Download Pythagoras` theorem and surds

Measurement and geometry
1
Pythagoras’
theorem
and surds
Pythagoras was an ancient Greek mathematician who lived
in the 5th century BCE. The theorem (or rule) which carries
his name was well-known before this time, but Pythagoras
may have been the first to prove it. Over 300 proofs for the
theorem are known today. Pythagoras’ theorem is perhaps
the most famous mathematical formula, and it is still used
today in architecture, engineering, surveying and astronomy.
N E W C E N T U R Y M AT H S A D V A N C E D
ustralian Curriculum
9
Shutterstock.com/Curioso
for the A
n Chapter outline
Proficiency strands
1-01 Finding the hypotenuse
U F
1-02 Finding a shorter side
U F
1-03 Surds and irrational numbers* U F
R C
1-04 Simplifying surds*
U F
R
1-05 Adding and subtracting surds* U F
R
1-06 Multiplying and dividing
surds*
U F
R
1-07 Pythagoras’ theorem
problems
F PS
C
1-08 Testing for right-angled
triangles
U F
R C
1-09 Pythagorean triads
U F
C
*STAGE 5.3
Pythagoras’ theorem is a Year 9 topic in the Australian Curriculum
but a Stage 4 topic in the NSW syllabus, so Pythagoras’ theorem
has also been covered in Chapter 1 of New Century Maths 8 for the
Australian Curriculum.
9780170193085
n Wordbank
converse A rule or statement turned back-to-front; the
reverse statement
hypotenuse The longest side of a right-angled triangle; the
side opposite the right angle
pffiffiffi
irrational number A number such as p or 2 that cannot
be expressed as a fraction
Pythagoras An ancient Greek mathematician who
discovered an important formula about the sides of a
right-angled triangle
Pythagorean triad A set of three numbers that follow
Pythagoras’ theorem, such as 3, 4, 5.
surd A square root (or other root) whose exact value
cannot be found
theorem Another name for a formal rule or formula
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
n In this chapter you will:
• investigate Pythagoras’ theorem and its application to solving simple problems involving rightangled triangles
• investigate irrational numbers and surds
• write answers to Pythagoras’ theorem problems in decimal or surd form
• (STAGE 5.3) simplify, add, subtract, multiply and divide surds
• test whether a triangle is right-angled
• investigate Pythagorean triads
SkillCheck
Worksheet
1
Evaluate each expression.
a 42
d 82 62
StartUp assignment 1
MAT09MGWK10001
2
2
b 10.3
pffiffiffiffiffi
49
e
c 3p2ffiffiffiffiffiffiffi
þ ffi5 2
f
121
Find the perimeter of each shape.
4 cm
b
c
9
27 mm
cm
a
12
cm
7 cm
11 mm
3
Select the square numbers from the following list of numbers.
44
81
25
100
75
72
16
50
64
32
Worksheet
Pythagoras’ discovery
MAT09MGWK10002
1-01 Finding the hypotenuse
Worksheet
A page of right-angled
triangles
Summary
MAT09MGWK10003
Worksheet
Pythagoras 1
MAT09MGWK00055
Skillsheet
Pythagoras’ theorem
Pythagoras’ theorem
For any right-angled triangle, the square of the hypotenuse
is equal to the sum of the squares of the other two sides.
If c is the length of the hypotenuse, and a and b
are the lengths of the other two sides, then:
c
b
a
c2 ¼ a2 þ b2
MAT09MGSS10001
4
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Example
ustralian Curriculum
9
Technology
1
GeoGebra: Pythagoras’
theorem
N
Write Pythagoras’ theorem for this triangle.
MAT09MGTC00007
Solution
p
m
p is the hypotenuse, so p 2 ¼ m 2 þ n 2
OR
NM is the hypotenuse, so NM 2 ¼ NP 2 þ PM 2
Technology worksheet
Finding the hypotenuse
MAT09MGCT10006
P
M
n
Technology worksheet
Excel worksheet:
Pythagoras’ theorem
Example
2
MAT09MGCT00024
Technology worksheet
Find the value of c in this triangle.
9 cm
Excel spreadsheet:
Pythagoras’ theorem
Solution
MAT09MGCT00009
We want to find the length of the hypotenuse.
40 cm
c cm
Using Pythagoras’ theorem:
c 2 ¼ 9 2 þ 402
Skillsheet
Spreadsheets
MAT09NASS10027
¼ 1681
pffiffiffiffiffiffiffiffiffiffi
c ¼ 1681
Use square root to find c.
¼ 41
An answer of c ¼ 41 looks reasonable because:
• the hypotenuse is the longest side
• from the diagram, the hypotenuse looks a
little longer than the side that is 40 cm
Example
3
Video tutorial
Pythagoras’ theorem
Find the length of the cable supporting this flagpole:
a as a surd
b correct to one decimal place.
C
MAT09MGVT10001
pffiffiffiffiffiffiffiffi
b AC ¼ 160
¼ 12:6491 . . .
12:6 m
9780170193085
cab
a AC 2 ¼ 122 þ 42
¼ 160
pffiffiffiffiffiffiffiffi
AC ¼ 160 m
12 m
le
Solution
Thispisffiffi the answer as a surd
(in
form).
A
4m
B
From part a.
Rounded to one decimal place.
5
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Example
4
In nPQR, \ P ¼ 90°, PQ ¼ 25 cm and PR ¼ 32 cm. Sketch the triangle and find the
length of the hypotenuse, p, correct to one decimal place.
Solution
p 2 ¼ 322 þ 252
¼ 1649
pffiffiffiffiffiffiffiffiffiffi
p ¼ 1649
¼ 40:607 881 01 . . .
Q
p cm
25 cm
40:6
[ The length of the hypotenuse is 40.6 cm.
Exercise 1-01
See Example 1
1
P
R
32 cm
Finding the hypotenuse
Write Pythagoras’ theorem for each right-angled triangle.
a
b C
N
c
X
w
d
p
k
m
x
D
M
K
c
t
C
e
H
f
f
D
z
k
h
W
P
K
n
d
y
Y
E
r
d
q
A
T
See Example 2
2
Find the length of the hypotenuse in each triangle.
B
b
a
8 cm
C
6 cm
B
c
18 cm
P
Q
5m
C
A
6
F
B
c
12 m
A
24 cm
R
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
T
d
35 cm
X
Y
R
f
e
24 mm
P
16 cm
W
Z
P
10 mm
g
h
R
H
4.5 m
i
K
4m
k
B
F
l
60 mm
I
H
20 m
A
24 cm
A
j
7 cm
C
4.2 m
T
V
J
L
16 cm
12 cm
18 m
11 mm
J
D
B
3
30 cm
K
12 cm
C
9
7.5 m
L
Find the length of the hypotenuse in each triangle, as a surd.
R
X
a
See Example 3
b 11 cm
c
L
54 mm
10 cm
P
Z
Y
15 cm
25 mm
K
N
16 cm
Q
d
D
6 cm
e V
34 mm
C
57 mm
f
Z
51 mm
9 cm
39 mm
E
T
4
B
A
C
Find the length of the hypotenuse in each triangle, correct to one decimal place.
Q
a
1.85 m
41 mm
N
c Z
67 mm
G
R
72 mm
d
1.76 m
R
b
Q
T
e
P
T
2.4 m
L
f
R
V
84 mm
49 mm
T
19 cm
6 m
F
24 cm
70 mm
V
H
9780170193085
V
7
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
5
A rectangular field is 100 m long and 50 m wide. How far
is it from one corner to the opposite corner, along
the diagonal? Select the correct answer A, B, C or D.
A 150 m
C 100.2 m
100 m
?
50 m
B 111.8 m
D 98.3 m
A firefighter places a ladder on a window sill 4.5 m above the
ground. If the foot of the ladder is 1.6 m from the wall,
how long is the ladder? Leave your answer as a surd.
7
a In n ABC, \ABC ¼ 90°, AB ¼ 39 cm and BC ¼ 57 cm. Find AC correct to one decimal
place.
Shutterstock.com/Sergey Ryzhov
6
See Example 4
b In n MPQ, \ PQM ¼ 90°, QM ¼ 2.4 m and PQ ¼ 3.7 m. Find PM correct to one decimal
place.
c In n RVJ, \ J ¼ 90°, JV ¼ 12.7 cm and JR ¼ 4.2 cm. Find RV correct to the nearest
millimetre.
d In n EGB, EG ¼ EB ¼ 127 mm and \ GEB ¼ 90°. Find the length of BG correct to one
decimal place.
Worksheet
Finding an unknown
side
e In n VZX, \ V ¼ 90°, VX ¼ 247 cm and VZ ¼ 3.6 m. Find ZX in metres, correct to the
nearest 0.1 m.
f In n PQR, \RPQ ¼ 90°, PQ ¼ 2.35 m and PR ¼ 5.8 m. Find QR in metres, correct to two
decimal places.
MAT09MGWK10004
Homework sheet
Pythagoras’ theorem 1
1-02 Finding a shorter side
MAT09MGHS10029
Video tutorial
Pythagoras’ theorem can also be used to find the length of a shorter side of a right-angled triangle,
if the hypotenuse and the other side are known.
Pythagoras’ theorem
MAT09MGVT10001
Technology worksheet
Excel worksheet:
Pythagoras’ theorem
MAT09MGCT00024
Technology worksheet
Excel spreadsheet:
Pythagoras’ theorem
MAT09MGCT00009
8
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Example
ustralian Curriculum
5
9
Puzzle sheet
Pythagoras 1
6 mm
Find the value of d in this triangle.
Puzzle sheet
d mm
Solution
We want to find the length of a shorter side.
Using Pythagoras’ theorem:
MAT09MGPS00039
10
mm
Pythagoras 2
MAT09MGPS00040
10 2 ¼ d 2 þ 62
100 ¼ d 2 þ 36
d 2 þ 36 ¼ 100
d 2 ¼ 100 36
¼ 64
pffiffiffiffiffi
d ¼ 64
¼8
From the diagram, a length of 8 mm looks reasonable because it must be shorter than the
hypotenuse, which is 10 mm.
Example
6
Find the value of x as a surd for this triangle.
8m
Solution
2
3m
2
2
8 ¼x þ3
xm
2
64 ¼ x þ 9
2
x þ 9 ¼ 64
x 2 ¼ 64 9
¼ 55
pffiffiffiffiffi
x ¼ 55
Leave the answer as a surd.
Exercise 1-02
Find the value of the pronumeral in each triangle.
c
cm
m
34
17
y cm
m
b
See Example 5
30 cm
15 m
x mm
a
5 mm
13 m
1
Finding a shorter side
ym
9780170193085
9
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
e
m
25 m
f
m
24 c
20
mm
d cm
30 c
m
m
xm
am
m
9m
15
mm
d
m
2
Find the value of the pronumeral in each triangle correct to one decimal place.
a
16 cm
b
27 cm
x cm
75 cm
c
x cm
x cm
7 cm
120 cm
20 cm
d
y cm
e
25 cm
32 m
58 m
f
32 m
21 m
am
43 cm
am
See Example 6
3
Find the value of the pronumeral in each triangle as a surd.
c
b
a
127 m
45 cm
e cm
gm
62 m
103 m
50 m
84 cm
xm
d
e
f
1.9 m
wm
3.7 cm
4.9 cm
4.2 m
am
67 m
204 m
p cm
4
Find the value of x in this rectangle. Select the
correct answer A, B, C or D.
pffiffiffiffiffi
pffiffiffiffiffi
A p20
B p80
ffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffi
C 136
D 208
A
12 m
B
5
Find the value of p in this rectangle. Select the
correct answer A, B, C or D.
A 40
C 32
10
B 36
D 28
D
p mm
xm
C
8m
105 mm
111 mm
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
6
In this diagram, O is the centre of a circle.
A perpendicular line is drawn from O to B such
that OB ’ AC and AB ¼ BC. Calculate the
length of OB. Select the correct answer
A, B, C or D.
A 1.3 m
C 1.9 m
ustralian Curriculum
150 cm
B
A
C
9
Worked solutions
Finding a shorter side
MAT09MGWS10001
?
B 1.8 m
D 3.4 m
2m
O
A square has a diagonal of length 30 cm. What is the length of each side of the square, correct
to the nearest millimetre? [Hint: Draw a diagram]
12
cm
h cm
8
cm
An equilateral triangle has sides of length 12 cm.
Find the perpendicular height, h, of the triangle,
correct to two decimal places.
12
7
1-03 Surds and irrational numbers
•
•
pffiffiffiffiffi
p25
ffiffiffiffiffi ¼ 5
81 ¼ 9
because 5 2 ¼ 25
because 9 2 ¼ 81
‘the square root of 25’
‘the square root of 81’
Most
pffiffiffi square roots do not give exact answers like the ones above. For example,
7 ¼ 2:645751311 . . . 2:6. Such roots are called surds.
pffiffi
pffiffi
3
A surd is a square root
, cube root
, or any type of root whose exact decimal or fraction
value cannot be found. As a decimal, its digits run endlessly without repeating (like p), so they are
neither terminating nor recurring decimals.
Rational numbers such as fractions, decimals and percentages, can be expressed in the form a
b
where a and b are integers (and b 6¼ 0), but surds are irrational numbers because they cannot
be expressed in this form.
9780170193085
11
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Example
7
Select the surds from this list of square roots:
pffiffiffiffiffi
72
pffiffiffiffiffiffiffiffi
121
pffiffiffiffiffi
64
pffiffiffiffiffi
90
pffiffiffiffiffi
28
Solution
pffiffiffiffiffi
72 ¼ 8:4852 . . .
pffiffiffiffiffiffiffiffi
121 ¼ 11
pffiffiffiffiffi
64 ¼ 8
pffiffiffiffiffi
90 ¼ 9:4868 . . .
pffiffiffiffiffi
28 ¼ 5:2915 . . .
pffiffiffiffiffi
pffiffiffiffiffi pffiffiffiffiffi
so the surds are 72, 90 and 28.
Example
8
Is each number rational or irrational?
pffiffiffiffiffiffiffi
pffiffiffi
b 3 8
c 7
a 42
5
d 0:6_
e 5p
Solution
a 4 2 ¼ 22
5
5
[ 4 2 is a rational number.
5
p
ffiffiffiffiffiffi
ffi
b 3 8 ¼ 2
pffiffiffiffiffiffiffi
[ 3 8 is a rational number.
pffiffiffi
c 7 ¼ 2:645 751 311 . . .
pffiffiffi
[ 7 is an irrational number.
d 0:6 ¼ 0:666 . . .
2
¼
3
_
[ 0:6 is a rational number.
e 5p ¼ 15.707 963 27…
which is in the form of a fraction a
b
which can be written as 2
1
The digits run endlessly without repeating.
which is a recurring decimal
which is a fraction
The digits run endlessly without repeating.
[ 5p is an irrational number.
12
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
Surds on a number line
9
Stage 5.3
The rational and irrational numbers together make up the real numbers. Any real number can be
represented by a point on the number line.
3
– 35_
– 10
–3
–2
–1
pffiffiffiffiffi
3 10 2:1544 . . .
3 ¼ 0:6
5
2 0:6666 . . .
3
120% ¼ 1.2
pffiffiffi
5 2:2360 . . .
0
120%
1
5
2
π
3
4
irrational (surd)
rational (fraction)
rational (fraction)
rational (percentage)
irrational (surd)
p 3.1415…
Example
2_
3
irrational (pi)
9
Use a pair of compasses and Pythagoras’ theorem to estimate the value of
line.
pffiffiffi
2 on a number
Worksheet
Surds on the
number line
MAT09MGWK10006
Solution
Step 1: Using a scale of 1 unit to 2 cm, draw a number line as shown.
1
0
2
3
Step 2: Construct a right-angled triangle on the number line withpbase
ffiffiffi length and height 1
unit as shown. By Pythagoras’ theorem, show that XZ ¼ 2 units.
Z
2
1
X
0
1
1
2
3
2
3
pffiffiffi
Step 3: With 0 as centre, use compasses with radius XZ 2 to draw anparc
ffiffiffi to meet the
number line at A as shown. The point A represents the value of 2 and should be
approximately 1.4142…
Z
2
1
X
0
9780170193085
1
1
A
13
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Exercise 1-03
See Example 7
See Example 8
Surds and irrational numbers
1
Which one of the following is a surd? Select the correct answer A, B, C or D.
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
pffiffiffi
pffiffiffiffiffiffiffiffi
A 9
B 225
C 160
D 81
2
Which one of the following is NOT a surd? Select the correct answer A, B, C or D.
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
B 144
C 18
D 200
A 77
3
Select the surds from the following list of square roots.
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
32
33
289
81
4:9
52
4
Is each number rational (R) or irrational (I)?
pffiffiffi
pffiffiffi
4
8
b
c
d 31
a 5:6_
7
pffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
e 3 27
f 1:35_
g 3 64
h 27 1 %
2
pffiffiffiffiffi
pp
ffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
50
3
3
4
5 3 10
i
j
k
l
11
3
Arrange each set of numbers in descending order.
pffiffiffi
pffiffiffiffiffi
_ 27
a 1 4, 2, p
b 3 20, 2:6,
7
2
9
pffiffiffi
Use the method from Example 9 to estimate the value of 2 on a number line.
pffiffiffi
a Use the method from Example 9 to estimate the value of 5 on a number line by
constructing a right-angled triangle with base length 2 units and height 1 unit.
5
Stage 5.3
6
See Example 9
7
pffiffiffiffiffiffiffiffi
121
pffiffiffiffiffiffiffiffi
144
pffiffiffiffiffiffiffiffi
196
pffiffiffiffiffiffiffiffi
200
b Use a similar method to estimate the following surds on a number line.
pffiffiffiffiffi
pffiffiffiffiffi
17
ii
10
i
Investigation: Proof that
pffiffiffi
2 is irrational
A method of proof sometimes used in mathematics is to assume the opposite of what is
being proved, and to show that p
it ffiffiisffi impossible. This is called a proof by contradiction,
2 is irrational.
and we will use it to prove
that
pffiffiffi
pffiffiffi
First, we assume that 2 is rational. So, assume that 2 can be written as a simplified
fraction a, where a and b are integers (b 6¼ 0) with no common factors.
b
pffiffiffi a
2¼
b
2
2 ¼ a 2 Squaring both sides
b
a2 ¼ 2b2
2b 2 is an even number because it is divisible by 2, [ a 2 is even.
[ a is even, because an even integer multiplied by itself is always even and an odd integer
multiplied by itself is always odd.
[ a ¼ 2m, where m is another integer.
[ a 2 ¼ (2m)2 ¼ 2b 2
4m2 ¼ 2b 2
2m2 ¼ b 2
b 2 ¼ 2m2
[ b 2 is even
[ b is even.
14
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
But a and b can’t both be even because this contradicts
pffiffiffi the assumption that apand
ffiffiffi b have
no common factors. Therefore, the assumption that 2 is rational is false, so 2 must be
irrational.
1 Usepproof
by contradiction p
toffiffiffishow that these surds are irrational:
ffiffiffi
a
b
3
5
2 Compare your proofs with those of other students.
Just for the record
Pythagoras and the Pythagoreans
Pythagoras was a mathematician who lived in ancient Greece.
The Pythagoreans were a group of men who followed
Pythagoras.
Sometimes when applying Pythagoras’ theorem, lengths are
found that cannot be expressed as exact rational numbers.
Pythagoras encountered this when calculating the diagonal of
a square of side length 1 unit.
The Pythagoreans were the first to study the properties of whole
numbers. They explained nature, the universe — in fact
everything — in terms of whole numbers. Apparently they were so
upset about the discovery of surds that they tried to keep the discovery a secret. Hippasus,
one of the Pythagoreans, was drowned for revealing the secret to outsiders.
p is an irrational number but it is not a surd. Why? Another such number is e. Investigate
the numbers p and e and the meaning of transcendental numbers.
1-04 Simplifying surds
The square of any real number is always positive (except for 0 2 ¼ 0), so it is not possible to give
the square root of a negative
number.
pffiffi
pffiffiffi
stands for the positive square root of a number, for example 4 ¼ 2
The radical symbol
(not 2).
Stage 5.3
Worksheet
Surds
MAT09NAWK10005
Puzzle sheet
Simplifying surds
MAT09NAPS10007
Summary
pffiffiffi
For x < 0 (negative), x is undefined.
pffiffiffi
For x ¼ 0, x is 0.
pffiffiffi
For x > 0 (positive), x is the positive square root of x.
pffiffiffi2 pffiffiffi pffiffiffi
x ¼ x3 x¼x
pffiffiffiffiffi
x2 ¼ x
9780170193085
15
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Stage 5.3
Example
10
Simplify each expression.
pffiffiffi2
pffiffiffi2
7
b 3 5
a
c
pffiffiffi2
2 3
Solution
pffiffiffi2
7 ¼7
pffiffiffi2
pffiffiffi
pffiffiffi
b 3 5 ¼ 3 533 5
pffiffiffi2
¼ 32 3 5
¼ 935
a
c
pffiffiffi
pffiffiffi
3 5 means 3 3 5
¼ 45
pffiffiffi2
pffiffiffi2
2 3 ¼ ð2Þ2 3 3
¼ 433
¼ 12
Summary
The square root of a product
For x > 0 and y > 0:
pffiffiffiffiffi pffiffiffi pffiffiffi
xy ¼ x 3 y
pffiffiffi
A surd n can be simplified if n can be divided into two factors where one of them is a square
number such as 4, 9, 16, 25, 36, 49, …
Example
11
Simplify each surd.
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
b
a
50
432
pffiffiffiffiffi
c 4 12
pffiffiffiffiffiffiffiffi
288
d
3
Solution
a
16
pffiffiffiffiffi pffiffiffiffiffi pffiffiffi
50 ¼ 25 3 2
pffiffiffi
¼ 53 2
pffiffiffi
¼5 2
25 is a square number.
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
b Method 1
pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi
432 ¼ 4 3 108
pffiffiffiffiffiffiffiffi
¼ 2 3 108
pffiffiffiffiffi pffiffiffi
¼ 2 3 36 3 3
pffiffiffi
¼ 2363 3
pffiffiffi
¼ 12 3 3
pffiffiffi
¼ 12 3
ustralian Curriculum
9
Stage 5.3
Method 2
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi
432 ¼ 144 3 3
pffiffiffi
¼ 12 3 3
pffiffiffi
¼ 12 3
pffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
Method 1 involves simplifying surds twice ( 432 and 108).
Method 2 shows that when simplifying surds, it is more efficient
to first look for the highest square factor possible.
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi
pffiffiffiffiffi
pffiffiffi pffiffiffi
144 3 2
288
c 4 12 ¼ 4 3 4 3 3
d
¼
3
3
pffiffiffi
pffiffiffi
¼ 4323 3
12 2
pffiffiffi
¼
3
¼8 3
4 pffiffiffi
12 2
¼
31
pffiffiffi
¼4 2
Exercise 1-04
1
2
Simplifying surds
Simplify each expression.
pffiffiffi2
pffiffiffi2
2
b
5
a
pffiffiffiffiffiffiffiffiffi2
pffiffiffi2
0:09
e
f 2 7
Simplify each surd.
pffiffiffi
a p8
ffiffiffiffiffiffiffiffi
e p243
ffiffiffiffiffi
i p96
ffiffiffiffiffi
m p75
ffiffiffiffiffiffiffiffi
q
162
b
f
j
n
r
pffiffiffiffiffi
p27
ffiffiffiffiffi
p45
ffiffiffiffiffi
p63
ffiffiffiffiffiffiffiffi
p147
ffiffiffiffiffiffiffiffi
245
pffiffiffiffiffi2
5 10
pffiffiffi2
h 5 2
pffiffiffi2
3 3
pffiffiffi2
g 3 5
d
pffiffiffiffiffi
p24
ffiffiffiffiffi
p48
ffiffiffiffiffiffiffiffi
p288
ffiffiffiffiffi
p32
ffiffiffiffiffiffiffiffi
125
d
h
l
p
t
c
c
g
k
o
s
pffiffiffiffiffi
p54
ffiffiffiffiffiffiffiffi
p200
ffiffiffiffiffiffiffiffi
p108
ffiffiffiffiffiffiffiffi
p242
ffiffiffiffiffiffiffiffi
512
3
Simplify each expression.
pffiffiffi
pffiffiffiffiffi
a 5 50
b 3 8
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
243
40
e
f
2
9
p
ffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
3125
i 9 68
j
10
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
m 10 160
n 3 75
4
Decide whether each statement is true (T) or false (F).
pffiffiffiffiffi
pffiffiffi pffiffiffiffiffi
pffiffiffiffiffiffiffi2
a 3 7 ¼ 21
9:4 ¼ 9:4
b
c
12 ¼ 6
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffi
pffiffiffi
d
e
f The exact value of 10 is 3.162 277 8
75 ¼ 5 3
3 1:7
9780170193085
pffiffiffiffiffi
c 4 27
pffiffiffiffiffi
28
g
6
pffiffiffiffiffi
k 1 72
2
pffiffiffiffiffi
o 7 68
See Example 10
See Example 11
pffiffiffiffiffi
d 8 98
pffiffiffiffiffi
h 3 24
pffiffiffiffiffi
l 3 48
4
pffiffiffiffiffi
52
p
6
17
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Mental skills 1A
Maths without calculators
Multiplying and dividing by a power of 10
Multiplying a number by 10, 100, 1000, etc. moves the decimal point to the right and
makes the number bigger. We place zeros at the end of the number if necessary.
•
•
•
When multiplying a number by 10, move the decimal point one place to the right.
When multiplying a number by 100, move the decimal point two places to the right.
When multiplying a number by 1000, move the decimal point three places to the right.
The number of places the decimal is moved to the right matches the number of zeros in
the 10, 100 or 1000 we are multiplying by.
1
2
Study each example.
a 26.32 × 10 = 26.32 = 263.2
The point moves one place to the right.
b 8.701 × 100 = 8.701 = 870.1
The point moves two places to the right.
c 6.01 × 1000 = 6.010 = 6010
The point moves three places to the right after
a 0 is placed at the end.
d 17 × 100 = 17.00 = 1700
The point moves two places to the right after
two zeros are placed at the end.
Now evaluate each expression.
a
d
g
j
89.54 3 10
42 3 100
31.84 3 1000
4.894 3 10
b
e
h
k
3.7 3 10
5.2716 3 1000
64.3 3 100
7.389 3 1000
c
f
i
l
0.831 3 100
156.1 3 10
0.0224 3 1000
11.42 3 100
Dividing a number by 10, 100, 1000, etc. moves the decimal point to the left and makes the
number smaller. We place zeros at the start of the decimal if necessary.
•
•
•
When dividing a number by 10, move the decimal point one place to the left.
When dividing a number by 100, move the decimal point two places to the left.
When dividing a number by 1000, move the decimal point three places to the left.
3
Study each example.
4
a 145.66 ÷ 10 = 145.66 = 14.566
The point moves one place to the left.
b 2.357 ÷ 100 = 002.357 = 0.023 57
The point moves two places to the left
after two zeros are inserted at the start.
c 14.9 ÷ 1000 = 0014.9 = 0.0149
The point moves three places to the left
after two zeros are inserted at the start.
d 45 ÷ 100 = 045. = 0.45
The point moves two places to the left
after one zero is inserted at the start.
Now evaluate each expression.
a
d
g
j
18
733.4 4 10
10.4 4 100
2 4 100
0.758 4 100
b
e
h
k
9.4 4 10
704 4 1000
4159 4 1000
8.49 4 100
c
f
i
l
652 4 100
198.5 4 100
123 4 10
25.1 4 1000
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
Investigation: A formula for calculating square roots
Calculators and computers use a formula repeatedly to give an approximate decimal
pffiffiffiffiffi
answer for the square root. The formula for calculating M is xnþ1 ¼ 1 xn þ M
xn where
2
x0 is the first guess and each calculation of the formula gives a better approximation than
the last one.
pffiffiffiffiffiffiffiffi
For example, to calculate 500 using a first guess of x0 ¼ 20:
x1 ¼ 1 20 þ 500 ¼ 22:5
2
20
1
x2 ¼
22:5 þ 500 ¼ 22:361 111 11
2
22:5
1
x3 ¼
22:36 þ 500 ¼ 22:360 679 78
2
22:36
This
process
can continue endlessly, with the accuracy increasing each time.
pffiffiffiffiffiffiffi
ffi
) 500 22:36 correct to two decimal places.
pffiffiffiffiffi
1 a Estimate 55.
pffiffiffiffiffi
b Use the formula
to
evaluate
55 to three decimal places, using the iterative formula.
pffiffiffiffiffi
c Evaluate 55 on your calculator to check your answer.
pffiffiffiffiffiffiffiffi
2 a Estimate 700.
pffiffiffiffiffiffiffiffi
b Use the formula
pffiffiffiffiffiffiffiffi to evaluate 700 to four decimal places.
c Evaluate 700 on your calculator to check your answer.
3 Design a spreadsheet that uses the formula repeatedly to calculate square roots.
1-05 Adding and subtracting surds
Stage 5.3
Puzzle sheet
Just as you can only add or subtract ‘like terms’ in algebra, you can only add or subtract ‘like
surds’.
Example
Surds code puzzle
MAT09MGPS10008
12
Simplify each expression.
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
b 7 32 3
a 4 2þ5 2
pffiffiffi pffiffiffiffiffi pffiffiffiffiffi
pffiffiffiffiffi pffiffiffiffiffi
d
e
8 27 þ 18
50 þ 32
pffiffiffi
pffiffiffi pffiffiffi
c 5 23 3þ 2
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
f 5 20 3 125
Solution
pffiffiffi
pffiffiffi
pffiffiffi
a 4 2þ5 2¼9 2
pffiffiffi
pffiffiffi
pffiffiffi
b 7 32 3¼5 3
pffiffiffi
pffiffiffi
pffiffiffi pffiffiffi
pffiffiffi
c 5 23 3þ 2¼6 23 3
9780170193085
19
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffipffiffiffi pffiffiffiffiffipffiffiffi
50 þ 32 ¼ 25 2 þ 16 2
pffiffiffi
pffiffiffi
¼5 2þ4 2
pffiffiffi
¼9 2
pffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffipffiffiffi pffiffiffipffiffiffi pffiffiffipffiffiffi
e
8 27 þ 18 ¼ 4 2 9 3 þ 9 2
pffiffiffi
pffiffiffi
pffiffiffi
¼2 23 3þ3 2
pffiffiffi
pffiffiffi
¼5 23 3
pffiffiffiffiffi
pffiffiffipffiffiffi
pffiffiffiffiffiffiffiffi
pffiffiffiffiffipffiffiffi
f 5 20 3 125 ¼ 5 4 5 3 25 5
pffiffiffi
pffiffiffi
¼ 532 5 335 5
pffiffiffi
pffiffiffi
¼ 10 5 15 5
pffiffiffi
¼ 5 5
d
Stage 5.3
Exercise 1-05
See Example 12
Adding and subtracting surds
1
Simplify each expression.
pffiffiffi
pffiffiffi
a 5 7þ2 7
pffiffiffi
pffiffiffi
5þ3 5
d
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
g 4 15 3 15 þ 7 15
pffiffiffi
pffiffiffi pffiffiffi
j 4 5þ7 5 5
2
Simplify each expression.
pffiffiffi
pffiffiffi
a 3 58þ2 5
pffiffiffi
pffiffiffi
pffiffiffi
c 4 3 þ 5 2 5 3
pffiffiffi
pffiffiffi pffiffiffi
pffiffiffi
73 54 7þ 5
e
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffi
pffiffiffi
g 10 11 5 3 þ 3 11 þ 4 3
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
i 2 53 72 53 7
3
Forpeach
expression,
select the correct simplified answer A, B, C or D.
ffiffiffiffiffi
ffiffiffi p
3 þ 12
a
pffiffiffi
pffiffiffi
pffiffiffiffiffi
A 5 3
B 15
C 2 6
pffiffiffi
pffiffiffiffiffiffiffiffi
b 4 5 2 125
pffiffiffi
pffiffiffiffiffi
pffiffiffi
A 6 5
B 5
C 45
4
20
Simplifying each surd.
Simplify each expression.
pffiffiffi pffiffiffiffiffi
8 þ 32
a
pffiffiffiffiffi pffiffiffiffiffi
28 63
d
pffiffiffiffiffi pffiffiffiffiffi
40 90
g
pffiffiffiffiffi
pffiffiffi
27 þ 5 3
j
pffiffiffiffiffi
pffiffiffi
m 5 3 þ 2 27
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
p 4 27 þ 2 243
pffiffiffi
pffiffiffiffiffiffiffiffi
s 5 6 þ 2 150
pffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
v 3 112 2 252
pffiffiffiffiffi
pffiffiffi
pffiffiffiffiffi
98 3 20 2 8
y
pffiffiffi
pffiffiffi
b 3 28 2
pffiffiffiffiffi
pffiffiffiffiffi
e 5 17 5 17
pffiffiffi
pffiffiffi
pffiffiffi
h 5 62 64 6
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
k 8 10 5 10 þ 3 10
pffiffiffi pffiffiffi
c 7 5 5
pffiffiffiffiffi
pffiffiffiffiffi
f 3 10 2 10
pffiffiffi
pffiffiffi
pffiffiffi
i 3 3þ4 35 3
pffiffiffi
pffiffiffi
pffiffiffi
l 10 3 3 3 12 3
pffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
b 11 10 þ 3 2 þ 2 10
pffiffiffi
pffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
d 3 15 þ 3 2 þ 4 15 þ 5 2
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
f 4 63 32 65 3
pffiffiffi
pffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
h
13 þ 8 7 7 13 þ 3 7
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffi
j 4 10 3 5 4 10
pffiffiffiffiffiffiffiffi pffiffiffiffiffi
b
108 27
pffiffiffi pffiffiffiffiffi
e 3 6 þ 24
pffiffiffiffiffi pffiffiffiffiffi
h 5 11 þ 99
pffiffiffi
pffiffiffiffiffiffiffiffi
k
200 7 2
pffiffiffiffiffi pffiffiffiffiffiffiffiffi
n 3 20 245
pffiffiffiffiffi
pffiffiffiffiffi
q 3 63 2 28
pffiffiffiffiffi
pffiffiffiffiffi
t 4 50 þ 3 18
pffiffiffiffiffi pffiffiffi pffiffiffiffiffi
w 32 þ 8 þ 12
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi pffiffiffiffiffi
z 3 96 2 150 þ 24
pffiffiffi
D 3 3
pffiffiffi
D 46 5
pffiffiffiffiffi pffiffiffiffiffi
c
20 80
pffiffiffi pffiffiffiffiffiffiffiffi
f 2 5 þ 125
pffiffiffi pffiffiffiffiffi
i 3 2 þ 18
pffiffiffiffiffi pffiffiffiffiffi
l
50 þ 32
pffiffiffiffiffi
pffiffiffiffiffi
o 7 12 5 48
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
r 2 98 þ 3 162
pffiffiffiffiffi
pffiffiffiffiffi
u 5 27 6 75
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffi
27 þ 54 þ 243
x
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
1-06 Multiplying and dividing surds
9
Stage 5.3
Summary
The square root of products and quotients
For x > 0 and y > 0:
pffiffiffiffiffi pffiffiffi pffiffiffi
xy ¼ x 3 y
qffiffiffi pffiffixffi
x ¼ pffiffiffi
y
y
Example
13
Simplify each expression.
pffiffiffiffiffi pffiffiffi
pffiffiffi pffiffiffi
a
b
10 3 6
33 5
pffiffiffi
pffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
d 5 27 3 3 6
e
54 4 2
Solution
a
pffiffiffi pffiffiffi pffiffiffiffiffi
3 3 5 ¼ 15
pffiffiffi
pffiffiffi
pffiffiffi pffiffiffi
c 3 735 7 ¼ 3353 73 7
¼ 15 3 7
¼ 105
pffiffiffiffiffi
pffiffiffi
pffiffiffiffiffi
54
e
54 4 2 ¼ pffiffiffi
2
pffiffiffiffiffi
¼ 27
pffiffiffi pffiffiffi
¼ 93 3
pffiffiffi
¼ 3 3
9780170193085
pffiffiffi
pffiffiffi
c 3 735 7
pffiffiffiffiffi
15 32
pffiffiffi
f
5 8
pffiffiffiffiffi pffiffiffi pffiffiffiffiffi
10 3 6 ¼ 60
pffiffiffi pffiffiffiffiffi
¼ 4 3 15
pffiffiffiffiffi
¼ 2 15
pffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi pffiffiffi
d 5 27 3 3 6 ¼ 5 3 3 3 27 3 6
pffiffiffiffiffiffiffiffi
¼ 15 162
pffiffiffiffiffi pffiffiffi
¼ 15 3 81 3 2
pffiffiffi
¼ 15 3 9 2
pffiffiffi
¼ 135 2
pffiffiffiffiffi
pffiffiffi
f 15 32
pffiffiffi ¼ 3 4
5 8
¼ 332
¼6
b
21
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Stage 5.3
Example
14
pffiffiffi
pffiffiffiffiffi
5 2 3p4ffiffiffi 12
.
Simplify
10 8
Solution
pffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
5 2 3 4 12 20 24
pffiffiffi
pffiffiffi
¼
10 8
10 8
pffiffiffi
¼2 3
Exercise 1-06
See Example 13
1
2
See Example 14
22
3
Multiplying and dividing surds
Simplify each expression.
pffiffiffi pffiffiffi
a
73 2
pffiffiffiffiffi pffiffiffi
12 3 3
d
pffiffiffiffiffi
pffiffiffi
g 5 10 3 3 3
pffiffiffi
pffiffiffi
j 2 3 3 5 6
pffiffiffi
pffiffiffi
m 7 2 3 4 8
pffiffiffiffiffi
pffiffiffiffiffi
p 3 18 3 5 12
pffiffiffi
pffiffiffiffiffi
s 8 3 3 3 54
pffiffiffiffiffi
pffiffiffi
v 5 20 3 3 8
pffiffiffi pffiffiffi
b 53 7
pffiffiffiffiffi
pffiffiffi
e
10 3 5
pffiffiffi
pffiffiffi
h 2 7 3 5 3
pffiffiffi pffiffiffiffiffi
k 4 3 3 27
pffiffiffiffiffi
pffiffiffi
18 3 8 3
n
pffiffiffiffiffi
pffiffiffiffiffi
q 3 44 3 2 99
pffiffiffiffiffi pffiffiffiffiffi
t 8 32 3 27
pffiffiffiffiffi
pffiffiffiffiffi
w 7 18 3 3 24
Simplify each expression.
pffiffiffiffiffi pffiffiffi
15 4 3
a
pffiffiffi
pffiffiffiffiffi
b
18 4 6
pffiffiffiffiffi
pffiffiffiffiffi
d 10 54 4 5 27
pffiffiffi
pffiffiffiffiffi
g 2 24 4 4 6
pffiffiffiffiffi
20p10
ffiffiffi
j
4 5
pffiffiffiffiffi
m 12 14 4 6
pffiffiffiffiffi pffiffiffiffiffi
p 5 60 4 15
pffiffiffiffiffi
pffiffiffi
s 12 63 4 3 7
pffiffiffiffiffi
pffiffiffiffiffi
e 3 98 4 6 14
pffiffiffiffiffiffiffiffi
128
h pffiffiffi
2
pffiffiffi
pffiffiffiffiffi
k 36 24 4 9 8
pffiffiffi
3 2
n
12
pffiffiffi
pffiffiffi
q 6 843 2
pffiffiffiffiffi
8 50
t pffiffiffiffiffiffiffiffi
2 200
Simplify each expression.
pffiffiffi
pffiffiffi
3 534 2
pffiffiffiffiffi
a
3 40
pffiffiffi
4 5
pffiffiffiffiffi
d pffiffiffiffiffi
2 15 3 5 27
pffiffiffi
pffiffiffiffiffi
3 12p3
8
ffiffiffiffiffi 6
b
4 27
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
10 p686
3 ffiffiffiffiffi12
ffiffiffiffiffi 3p
e
5 28 3 18
c
f
i
l
o
r
u
x
c
f
i
l
o
r
u
pffiffiffi pffiffiffi
63 8
pffiffiffi
pffiffiffi
3 335 3
pffiffiffi
pffiffiffi
7 534 5
pffiffiffiffiffi
pffiffiffi
3 5 3 4 10
pffiffiffi
pffiffiffi
10 2 3 2 8
pffiffiffi
pffiffiffiffiffi
5 8 3 4 40
pffiffiffiffiffi pffiffiffiffiffi
90 3 72
pffiffiffiffiffi
pffiffiffiffiffi
3 48 3 2 42
pffiffiffiffiffi
6 p48
ffiffiffi
2 8
pffiffiffiffiffi
7p18
ffiffiffi
2
pffiffiffi
pffiffiffiffiffi
15 18 4 3 6
pffiffiffiffiffi
pffiffiffi
16 30 4 8 5
pffiffiffiffiffi
pffiffiffi
80 4 4 5
pffiffiffiffiffi
42 54
pffiffiffi
6 3
pffiffiffi pffiffiffiffiffiffiffiffi
6 3 4 243
pffiffiffi
pffiffiffiffiffi
5 83
2
pffiffiffiffiffi 90
c
10 24
pffiffiffi
pffiffiffiffiffi
8 80 3 3pffiffi2ffi
pffiffiffi
f
4 536 8
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
Worksheet
1-07 Pythagoras’ theorem problems
Applications of
Pythagoras’ theorem
When using Pythagoras’ theorem to solve problems, it is useful to follow these steps.
•
•
•
•
•
9
MAT09MGWK100009
Read the problem carefully
Draw a diagram involving a right-angled triangle and label any given information
Choose a variable to represent the length or distance you want to find
Use Pythagoras’ theorem to find the value of the variable
Answer the question
Example
15
Find the value of y as a surd.
12 cm
Solution
cm
y cm
28
y is the length of a shorter side.
282 ¼ y 2 þ 122
784 ¼ y 2 þ 144
y 2 þ 144 ¼ 784
y 2 ¼ 784 144
¼ 640
pffiffiffiffiffiffiffiffi
y ¼ 640
Example
16
A ship sails 80 kilometres south and then 45 kilometres east.
How far is it from its starting point, correct to one decimal place?
N
Solution
Let x be the distance the ship is from the starting point.
x 2 ¼ 80 2 þ 452
¼ 8425
pffiffiffiffiffiffiffiffiffiffi
x ¼ 8425
¼ 91:7877 . . .
91:8 km
9780170193085
80 km
From the diagram, this looks
like a reasonable answer
x
45 km
ship
23
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Example
17
Find the perimeter of this triangle, correct to one decimal place.
10 m
Solution
21 m
Let x be the length of the hypotenuse.
xm
x 2 ¼ 10 2 þ 212
¼ 541
pffiffiffiffiffiffiffiffi
x ¼ 541
23:3 m
Perimeter 10 þ 21 þ 23:3
¼ 57:3 m
Example
18
Find the value of y correct to two
decimal places.
B
y
15
C
12
A
13
D
Solution
We need to find BD first.
In nABD,
BD 2 ¼ 15 2 þ 132
¼ 394
pffiffiffiffiffiffiffiffi
BD ¼ 394
In nBCD,
pffiffiffiffiffiffiffiffi
y 2 ¼ ð 394Þ2 þ 122
¼ 538
pffiffiffiffiffiffiffiffi
y ¼ 538
¼ 23:1948 . . .
23:19
24
Leave BD as a surd (don’t round)
for further working.
pffiffiffiffiffiffiffiffi2
394 ¼ 394
From the diagram, this looks
like a reasonable answer.
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Exercise 1-07
ustralian Curriculum
Pythagoras’ theorem problems
1 Find the value of the pronumeral in each triangle. Give your answers correct to one
decimal place.
a
9
c
b
17 cm
m cm
4m
96 mm
72 mm
km
27 cm
See Example 15
g mm
2.4 m
p cm
ym
f
17 mm
32 mm
19.5 m
18.0 cm
34.1 m
x mm
5m
2 A ladder 5 m long is leaning against a wall. If the
base of the ladder is 2 m from the bottom of the wall,
how far does the ladder reach up the wall?
(Leave your answer as a surd.)
2m
3 The size of a television screen is described by
the length of its diagonal. If a flat screen TV is
58 cm wide and 32 cm high, what is the size
of the screen? Answer to the nearest centimetre.
58 cm
?
32 cm
4 A ship sails 70 kilometres west and then 60 kilometres north. How far is it from its starting
point, correct to one decimal place?
9780170193085
Shutterstock.com/Pakhnyushcha
21.6 cm
e
iStockphoto/bmcent1
d
See Example 16
25
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
5 For each triangle drawn on the number plane, use Pythagoras’ theorem to calculate the length
of the hypotenuse. In part b, write the answer as a surd.
a
y
5
4
(3, 4)
3
2
1
0
1
2
3
4
5 x
b
y
5
4
3
2
1
–5 –4 –3 –2 –1
–1
1
2
3
4
5
x
–2
–3
–4
–5
See Example 17
6 Calculate the perimeter of each shape, correct to one decimal place where necessary.
a
b
c
9
cm
65
cm
16 cm
13 cm
25 cm
12 cm
40 mm
5 cm
e
f
5c
m
6
1.
m
2.4 m
31 mm
20 mm
12 cm
d
1.4 m
26
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
7 a Find the value of x correct to 1 decimal place.
ustralian Curriculum
9
See Example 18
A
Worked solutions
5 cm
x cm
D
Mixed problems
MAT09MGWS10002
12 cm
B
36 cm
C
b Find the length of AD correct to
2 decimal places.
A
8
B
6
C
7
D
c Calculate the length of x as a surd.
7
x
8
12
9780170193085
X
450 m
Z
Y
780 m
Corbis/Ó Hubert Stadler
8 Cooper wanted to find the width XY of
Lake Hartzer shown on the right. He placed
a stake at Z so that \ YXZ ¼ 90°. He measured
XZ to be 450 m long and ZY to be 780 m long.
What is the width of the lake?
Answer correct to one decimal place.
27
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
9 The swim course of a triathlon race has the shape of a right-angled triangle joined with
a 2 metre link to the beach as shown below. Calculate the total distance covered in this
course (starting and finishing on the beach).
2m
START/FINISH
SWIM LEG
264 m
BEACH
Buoy 1
170 m
Buoy 2
10 A kite is attached to a 24 m piece of rope, as shown.
The rope is held 1.2 m above the ground and covers
a horizontal distance of 10 m.
Find:
a the value of x correct to one decimal place
b the height of the kite above the ground,
correct to the nearest metre.
Worksheet
Pythagorean triads
24 m
1.2 m
xm
10 m
1-08 Testing for right-angled triangles
MAT09MGWK00056
Pythagoras’ theorem can also be used to test whether
a triangle is right-angled.
Pythagoras’ theorem says that if a right-angled triangle
has sides of length a, b, and c, then c 2 ¼ a 2 þ b 2.
The reverse of this is also true.
c
b
a
If any triangle has sides of length a, b, and c that follow the formula c 2 ¼ a 2 þ b 2, then the
triangle must be right-angled. The right angle is always the angle that is opposite the hypotenuse.
This is called the converse (or opposite) of Pythagoras’ theorem, because it is the ‘back-to-front’
version of the theorem.
28
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Example
ustralian Curriculum
19
9
Video tutorial
Testing for right-angled
triangles
A
Test whether n ABC is right-angled.
MAT09MGVT10002
75 mm
C
21 mm
B
72 mm
Solution
752 ¼ 5625
2
Square the longest side.
2
Square the two shorter sides, then add.
21 þ 72 ¼ 5625
2
2
) 75 ¼ 21 þ 72
2
[ n ABC is right-angled.
Example
The sides of this triangle follow c 2 ¼ a 2 þ b 2
The right angle is \B.
20
Rahul constructed a triangle with sides of length 37 cm, 12 cm and 40 cm. Show that these
measurements do not form a right-angled triangle.
Solution
402 ¼ 1600
2
Square the longest side.
2
Square the two shorter sides, then add.
12 þ 37 ¼ 1513 6¼ 1600
2
2
2
) 40 6¼ 12 þ 37
[ The triangle is not right-angled.
Exercise 1-08
1
The sides of this triangle do not follow c 2 ¼ a 2 þ b 2
Testing for right-angled triangles
Test whether each triangle is right-angled.
See Example 19
b
a
12
5
c
40
26
10
13
42
9
24
e
d
15
f
24
6
17
8
25
80
82
18
9780170193085
29
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
g
2.5
h
i
7.1
8.5
1.5
2
8.5
12.1
9.8
7.1
j
k
18
7.5
3.2
19.5
See Example 20
l
12.5
1.9
3.5
2.2
12
2
Penelope constructed a triangle with sides of length 17 cm, 11 cm and 30 cm. Show that these
measurements do not form a right-angled triangle.
3
Which set of measurements would make a right-angled triangle? Select the correct answer
A, B, C or D.
A 2 cm, 3 cm, 4 cm
B 5 mm, 10 mm, 15 mm
C 12 cm, 16 cm, 20 cm
D 7 m, 24 m, 31 m
4
Which one of these triangles is not right-angled? Select A, B, C or D.
A
B
12
7
40
5
13
45
C
D
60
80
35
28
100
21
30
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Mental skills 1B
ustralian Curriculum
9
Maths without calculators
Multiplying and dividing by a multiple of 10
1
Consider each example.
a
b
c
d
e
f
2
4 3 700 ¼ 4 3 7 3 100 ¼ 28 3 100 ¼ 2800
5 3 60 ¼ 5 3 6 3 10 ¼ 30 3 10 ¼ 300
12 3 40 ¼ 12 3 4 3 10 ¼ 48 3 10 ¼ 480
3.2 3 30 ¼ 3.2 3 3 3 10 ¼ 9.6 3 10 ¼ 96
(by estimation, 3 3 30 ¼ 90 96)
4.6 3 50 ¼ 4.6 3 5 3 10 ¼ 23 3 10 ¼ 230
(by estimation, 5 3 50 ¼ 250 230)
(by estimation,
9.4 3 200 ¼ 9.4 3 2 3 100 ¼ 18.8 3 100 ¼ 1880
9 3 200 ¼ 1800 1880)
Now evaluate each product.
a 8 3 2000
e 4 3 4000
i 2.5 3 600
3
c 11 3 900
g 7 3 70
k 3.6 3 50
d 2 3 300
h 1.3 3 40
l 4.4 3 3000
Consider each example.
a
b
c
d
e
f
4
b 3 3 70
f 5 3 80
j 5.8 3 200
8000 4 400 ¼ 8000 4 100 4 4 ¼ 80 4 4 ¼ 20
200 4 50 ¼ 200 4 10 4 5 ¼ 20 4 5 ¼ 4
6000 4 20 ¼ 6000 4 10 4 2 ¼ 600 4 2 ¼ 300
282 4 30 ¼ 282 4 10 4 3 ¼ 28.2 4 3 ¼ 9.4
3520 4 40 ¼ 3520 4 10 4 4 ¼ 352 4 4 ¼ 88
8940 4 200 ¼ 8940 4 100 4 2 ¼ 89.4 4 2 ¼ 44.7
Now evaluate each quotient.
a 560 4 70
e 160 4 40
i 2550 4 300
9780170193085
b 2500 4 500
f 1500 4 30
j 846 4 200
c 3200 4 400
g 450 4 50
k 576 4 60
d 440 4 20
h 744 4 80
l 2160 4 90
31
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Worksheet
Pythagorean triads
1-09 Pythagorean triads
MAT09MGWK10010
Homework sheet
Pythagoras’ theorem 2
A Pythagorean triad or Pythagorean triple is any group of three numbers that follow Pythagoras’
theorem, for example, (3, 4, 5) or (2.5, 6, 6.5). The word triad means a group of three related
items (‘tri-’ means 3).
MAT09MGHS10030
Homework sheet
Pythagoras’ theorem
revision
Summary
(a, b, c) is a Pythagorean triad if c 2 ¼ a 2 þ b 2
MAT09MGHS10031
Any multiple of (a, b, c) is also a Pythagorean triad.
Technology worksheet
Excel worksheet:
Pythagorean triples
Example
21
Test whether (5, 12, 13) is a Pythagorean triad.
MAT09MGCT00025
Technology worksheet
Excel spreadsheet:
Pythagorean triples
MAT09MGCT00010
Solution
13 2 ¼ 169
2
Squaring the largest number.
2
5 þ 12 ¼ 169
2
Squaring the two smaller numbers, and adding.
2
[ 13 ¼ 5 þ 12
2
These three numbers follow Pythagoras’ theorem.
[ (5, 12, 13) is a Pythagorean triad.
Example
22
(3, 4, 5) is a Pythagorean triad. Create other Pythagorean triads by multiplying (3, 4, 5) by:
a 2
b 9
c 1
2
Solution
a 2 3 (3, 4, 5) ¼ (6, 8, 10)
Checking:
2
10 ¼ 100
6 2 þ 8 2 ¼ 100
[ 10 2 ¼ 6 2 þ 8 2
[ (6, 8, 10) is a Pythagorean triad.
b 9 3 (3, 4, 5) ¼ (27, 36, 45)
Checking:
45 2 ¼ 2025
27 2 þ 36 2 ¼ 2025
[ 45 2 ¼ 27 2 þ 36 2
[ (27, 36, 45) is a Pythagorean triad.
c 1 3 (3, 4, 5) ¼ (1.5, 2, 2.5)
2
Checking: 2.5 2 ¼ 6.25
1.5 2 þ 2 2 ¼ 6.25
[ 2.5 2 ¼ 1.5 2 þ 2 2
[ (1.5, 2, 2.5) is a Pythagorean triad.
32
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Exercise 1-09
1
3
4
9
Pythagorean triads
Test whether each triad is a Pythagorean triad.
a (8, 15, 17)
d (5, 7, 9)
g (11, 60, 61)
2
ustralian Curriculum
b (10, 24, 26)
e (9, 40, 41)
h (7, 24, 25)
See Example 21
c (30, 40, 50)
f (4, 5, 9)
i (15, 114, 115)
Which of the following is a Pythagorean triad? Select the correct answer A, B, C or D.
A (4, 6, 8)
B (5, 10, 12)
C (6, 7, 10)
D (20, 48, 52)
Technology worksheet
Use the spreadsheet you created in Technology worksheet: Finding the hypotenuse to check
your answers to questions 1 and 2.
Finding the hypotenuse
For each Pythagorean triad, create another Pythagorean triad by multiplying each number in
the triad by:
i a whole number
ii a fraction
iii a decimal.
See Example 22
a (5, 12, 13)
b (8, 15, 17)
c (30, 40, 50)
MAT09MGCT10006
d (7, 24, 25)
Check that each answer follows Pythagoras’ theorem.
5
Pythagoras developed a formula for finding Pythagorean triads (a, b, c). If one number in the
triad is a, the formulas for the other two numbers are b ¼ 1 ða 2 1Þ and c ¼ 1 ða 2 þ 1Þ.
2
2
a If a ¼ 5, use the formulas to find the values of b and c.
Worked solutions
Pythagorean triads
MAT09MGWS10003
b Show that (a, b, c) is a Pythagorean triad.
6
Use the formulas to find Pythagorean triads for each value of a.
a a¼7
b a ¼ 11
c a ¼ 15
d a¼4
e a¼9
f a ¼ 19
g a ¼ 10
h a ¼ 51
7
There are many other formulas for creating Pythagorean triads. Use the Internet to research
some of them.
9780170193085
33
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Pythagoras’ theorem and surds
Power plus
1
Name the five hypotenuses
on this diagram.
B
A
E
D
2
The rectangle ABDE in the diagram
below has an area of 160 m 2. Find
the value of x as a surd if n BDC is
an isosceles triangle.
20 m
A
Find the value of x in each triangle,
correct to one decimal place.
B
xm
E
3
C
F
C
D
x
a
x
b
15
32
x
x
E
F
C
B
5m
For this rectangular prism, find, correct to
one decimal place, the length of diagonal:
a HD
b DE
H
G
m
4
6
A
5
An interval is drawn between points A(2, 1) and B(3, 2) on the number plane. Find
the length of interval AB correct to one decimal place.
6
For this cube, find, correct to two decimal
places, the length of diagonal:
a QS
b QT
U
T
N
M
R
Q
34
D
15 m
15 cm
S
P
9780170193085
Chapter 1 review
n Language of maths
Puzzle sheet
area
converse
diagonal
formula
hypotenuse
irrational
perimeter
Pythagoras
right-angled
shorter
side
square root
surd
theorem
triad
unknown
Pythagoras’ theorem
find-a-word
MAT09MGPS10011
1 Who was Pythagoras and what country did he come from?
2 Describe the hypotenuse of a right-angled triangle in two ways.
3 What is another word for ‘theorem’?
4 For what type of triangle is Pythagoras’ theorem used?
5 What is a surd?
6 What is the name given to a set of three numbers that follows Pythagoras’ theorem?
n Topic overview
•
•
•
•
How relevant do you think Pythagoras’ theorem is to our world? Give reasons for your answer.
Give three examples of jobs where Pythagoras’ theorem would be used.
What did you find especially interesting about this topic?
Is there any section of this topic that you found difficult? Discuss any problems with your
teacher or a friend.
Copy (or print) and complete this mind map of the topic, adding detail to its branches and using
pictures, symbols and colour where needed. Ask your teacher to check your work.
Finding a
shorter side
Worksheet
Mind map: Pythagoras’
theorem and surds
(Advanced)
MAT09MGWK10013
SU
Simplifying
surds
S
RD
PYTHAGORAS’
D
N
9780170193085
A
Testing for
right-angled
triangles
M
RE
Pythagorean
triads
Surds
and
irrational
numbers
EO
TH
Finding the
hypotenuse
Pythagoras’
theorem
problems
Operations
with
surds
35
Chapter 1 revision
See Exercise 1-01
1 Write Pythagoras’ theorem for each triangle.
A
a
b
F
c
n
E
B
q
p
C
See Exercise 1-01
2 Find the value of y. Select the closest answer A, B, C or D.
A 12.3
B 9.1
C 6.9
D 3.5
8.1 cm
4.2 cm
See Exercise 1-01
G
m
yc
3 For each triangle, find the length of the hypotenuse. Give your answer as a surd where
necessary.
a
b
140 mm
c
m
15
m
7.6 m
112 mm
6.4 m
105 mm
See Exercise 1-02
4 Find the value of d. Select the closest answer A, B, C or D.
A 12 m
B 15 m
C 16.8 m
D 18.9 m
8 cm
d cm
17 cm
See Exercise 1-02
5 For each triangle, find the length of the unknown side. Give your answer correct to one
decimal place.
a
c
b
72 m
15 cm
360 m
63 cm
0.9 km
8.7 cm
Stage 5.3
See Exercise 1-03
See Exercise 1-04
36
6 Is each number rational (R) or irrational (I)?
pffiffiffi
8
b 22
c 0:57_
a
7
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
_ 3_
f 3 125
g 3 8
h 0:12
7 Simplify each surd.
pffiffiffiffiffi
pffiffiffiffiffi
72
98
b
a
pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
f 7 28
g 4 288
pffiffiffiffiffiffiffiffi
275
pffiffiffiffiffi
h 5 45
c
pffiffiffi
d 3 5
pffiffiffi
i 5þ 3
pffiffiffiffiffiffiffiffi
128
pffiffiffiffiffi
i 7 48
d
e
pffiffiffiffiffi
81
pffiffiffiffiffiffiffiffi
e 3 150
9780170193085
Chapter 1 revision
8 Simplify each expression.
pffiffiffi pffiffiffi
a 8 5 5
pffiffiffi
pffiffiffi
c 6þ 54þ5 5
pffiffiffiffiffiffiffiffi pffiffiffiffiffi
e
200 þ 18
pffiffiffiffiffi
pffiffiffiffiffi pffiffiffiffiffi
pffiffiffiffiffi
g 7 32 27 2 98 þ 4 75
9 Simplify each expression.
pffiffiffi pffiffiffi
a
b
33 7
pffiffiffi pffiffiffiffiffi
d
e
5 3 11
pffiffiffiffiffi
pffiffiffi
g 8 42 4 2 7
h
pffiffiffiffiffi pffiffiffi
18 3 3
pffiffiffiffiffi
j
k
12
b
d
f
h
Stage 5.3
pffiffiffi
pffiffiffi
pffiffiffi
73 2þ2 7
pffiffiffi pffiffiffi
pffiffiffi
7 7 73 7
pffiffiffi pffiffiffiffiffi
pffiffiffiffiffiffiffiffi
3 5 þ 50 2 125
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
4 45 3 63 þ 5 80
See Exercise 1-05
See Exercise 1-06
pffiffiffi pffiffiffi
c
63 8
pffiffiffiffiffi pffiffiffi
98 4 7
f
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
i 4 147 4 12 98
pffiffiffi pffiffiffi
83 5
pffiffiffi pffiffiffi
83 3
pffiffiffiffiffiffiffiffi
pffiffiffi
125 4 5 5
pffiffiffi pffiffiffiffiffi
6 3 24
pffiffiffiffiffi
pffiffiffi
27 3 2 3
10 In n DEF, \ F ¼ 90°, DF ¼ 84 cm, and EF ¼ 1.45 m. Find DE correct to the nearest
centimetre.
See Exercise 1-07
11 Julie walks 6 km due east from a starting point P, while Jane walks 4 km due south from P.
a Draw a diagram showing this information.
b How far are Julie and Jane apart? Give your answer correct to one decimal place.
See Exercise 1-07
12 On a cross-country ski course, checkpoints C and A
are 540 m apart and checkpoints C and B are 324 m
apart. Find the distance:
See Exercise 1-07
B
324 m
a between checkpoints A and B
b skied in one lap of the course in kilometres.
C
540 m
A
13 Find the length of QR in this trapezium, correct
to one decimal place.
M
47 cm
60 cm
Q
14 Test whether each triangle is right-angled.
b
See Exercise 1-08
c
37
69
51
45
35
115
12
92
20
15 Test whether each triad is a Pythagorean triad.
a (7, 24, 25)
9780170193085
See Exercise 1-07
72 cm
R
a
P
b (5, 7, 10)
c (20, 21, 29)
See Exercise 1-09
d (1.1, 6, 6.1)
37