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The following people have worked on this book: Editorial team: Lucía de la Rosa (adaptation of material from the book Matematicas 2.o ESO and creation of the fact files and activities in this book) DESIGN, TECHNICAL DRAWINGS AND MAPS: Miguel Ángel Castillejos, Miguel Ángel Díaz-Rullo, Marta Gómez, Patricia G. Serrano ILLUSTRATIONS: Celia López Layout: Elvira López and Celia López Translation: Robin Munby PICTURE EDITING: Olga Sayans Photographs: Age Fotostock (Dawna Moore), Archivo Anaya (Cosano, P.; Osuna, J.), iStock/Gettyimages (DrPAS, JackF, Nerthuz), 123rf (Olga Yastremska)
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Index 1. NATURAL NUMBERS AND INTEGERS........................ 7
8. SYSTEMS OF EQUATIONS.............................................65
1 The set of natural numbers • 2 The relation of divisibility • 3 Prime and composite numbers • 4 Lowest common multiple of two or more numbers • 5 Greatest common divisor of two or more numbers • 6 The º set of integers • 7 Operations with integers • 8 Powers of integers • 9 Roots of integers
1 First-degree equations with two unknowns • 2 Systems of linear equations • 3 Methods for solving linear system • 4 Solving problems with systems of equations
2. DECIMAL NUMBERS AND FRACTIONS....................17 1 Decimal numbers • 2 Operations with decimal numbers • 3 Decimal numbers and sexagesimal numbers • 4 The square root of a decimal number • 5 Fractions • 6 Fractions and decimal numbers
3. OPERATIONS WITH FRACTIONS..................................25 1 Adding and subtracting fractions • 2 Multiplying and dividing fractions • 3 Problems with fractions • 4 Powers and fractions
4. PROPORTIONALITY...............................................................33 1 Ratios and proportions • 2 Directly proportional quantities • 3 Inversely proportional quantities • 4 Problems of compound proportionality • 5 Problems of proportional distribution
5. PERCENTAGES.......................................................................... 41 1 Percentages. Concept • 2 Problems with percentages • 3 Bank interest • 4 Other arithmetic problems
19. PYTHAGORAS’ THEOREM................................................ 73 1 Pythagoras’ theorem • 2 Calculating a side when two are known • 3 Applications of Pythagoras’ theorem
10. SIMILARITY............................................................................79 1 Similar shapes • 2 Plans, maps and models • 3 How to build similar figures • 4 Thales’ theorem • 5 Similarity between right-angled triangles • 6 Applications of the similarity of triangles
11. Three-dimensional geometric shapes.87 1 Prisms • 2 Pyramids • 3 Truncated pyramids • 4 Regular polyhedra • 5 Cross-sections of polyhedra • 6 Cylinders • 7 Cones • 8 Truncated cones • 9 Spheres • 10 Sections of spheres, cylinders and cones
12. Measuring volume......................................................97 1 Units of volume • 2 Cavalieri’s principle • 3 Volume of a prism and a cylinder • 4 Volume of a pyramid and a truncated pyramid • 5 Volume of a cone and a truncated cone • 6 Volume of a sphere
13. Functions...........................................................................105
1 Why do we use algebra? • 2 Algebraic expressions • 3 Polynomials • 4 Notable products
1 The concept of a function • 2 Increases, decreases, maximums and minimums • 3 Functions given as tables of values • 4 Functions given as equations • 5 Proportional functions: y = mx • 6 The slope of a line • 7 Linear functions: y = mx + n • 8 Constant functions: y=k
7. EQUATIONS.................................................................................57
14. statistics............................................................................ 113
1 Equations: meaning and use • 2 Equations: elements and names • 3 Transposing terms • 4 Solving simple equations • 5 Equations with denominators • 6 The general method for solving first-degree equations • 7 Solving problems with equations • 8 Second-degree equations • 9 Solving second-degree equations
1 Drawing a table and its graph • 2 Location parameters • 3 Dispersion parameters • 4 Measures of position • 5 Two-way tables
6. ALGEBRA..................................................................................... 49
DUAL FOCUS
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k oo b r u o y f o t In this par ind... you will f
MATHEMATICS
MATHEMATICS
ION SECONDARY EDUCAT
2
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10
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SECONDARY EDUCATION
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cacion.es www.anayaedu
Scan this code to consult the glossary for this unit.
Similarity
Similarity ratio of similarity relationship between the areas of two similar shapes
similar shapes
This section contains activities for practising and applying the key vocabulary in the unit.
h h’
relationship between the volumes of two similar shapes plans, maps and models
Focus on English
scale
The Final Challenge!
the grid method how to build similar figures
projection method triangles in Thales’ position
Each unit ends with a task or challenge to help you develop your skills by applying your knowledge in a different context.
similarity of triangles rule on the similarity of triangles
Thales' theorem
similarity between right-angled triangles
h
Focus on English
leg rule
1
Vocabulary The page that shows the key vocabulary you will encounter in the unit.
height rule
Listening. Listen and repeat. The vocabulary is at anayaeducacion.es.
2 Discover. Thales of Miletus (7th century BCE) was one of the greatest astronomers and mathematicians of his time. In addition to various mathematical discoveries (including the theorem that bears his name), he is also known for predicting the eclipse that took place in 585 BCE and for being the first to determine the number of days in a year.
calculating the height of an object
3
THE F I N A L CHALLENGE
Reading. To learn more, read the text ‘Greek geometry’ at anayaeducacion.es and answer the
NK questions. BANK NGUAGE BA NK LABANK LANGUAGENK NGUAGE BA NK LA BANK LANGUAGENK LANGUAGE BA BA GE UA LANG UAGE BA NGUAGE BANK BANK LAUAGE BANK LABANGNK LANGUAGE LANGUAGE LANG LANGUAGE
INVESTIGATE 79
How to mark out a beach volleyball court We want to mark out a beach volleyball court. How do we draw the lines? The best way is with a rope pulled tight. How do you get the right angle in the corners? Take a rope and mark twelve identical sections by making knots. With three stakes, tighten the rope to form a triangle with 3, 4 and 5 knots on the sides. It is a right-angled triangle. The right angle is at the vertex where the sides with 3 and 4 knots meet. A bit of history Over 3 000 years ago, the Egyptians used this method to draw right angles.
Basic exercises
The river Nile would burst its banks and flood farmland every year, and the borders between the flooded fields needed to be restored. The land surveyors who were responsible for marking the borders again used the method described above.
The activities on these pages work on the basic unit content, as well as the different language skills.
• Two right-angled
triangles with an equal .
6
RIGHT-ANG LED TRIAN GLES acute angle are similar,
: height of the tree AB method to find out the We use the following A'B' . vertically into the ground of the • We hammer a stake we measure the length of the stake, A'B' . Then the Sun at • We measure the length stake, that are made by 'C' , of the tree and the A and AC s, shadow angles: the same moment. they have two equal A'B'C' are similar because Triangles ABC and are right angles. same angle. A = A' because both tree and the stake at the Sun’s rays shine on the C = C' because the ional: their sides are proport similar, are s triangle Since the AB = AC A'B' A'C' AB . e the height of the tree, ' and A'C' , we can calculat As we know AC , A'B
since they can
a
a
^
A
triangles are similar if: ional, ✓ the two legs are proport ional. hypotenuse are proport two triangles ✓ or one leg and the the hypotenuse makes triangle, the height of • In a right-angled . similar to the original
b
• Two right-angled
n
M
a
use and the projection
the hypoten equal to the product of The square of a leg is use. of that leg over the hypoten a = c → c2 = a · n a = b → b2 = a · m c n b m
1
c
h
m
C
B
b m
C
PYTH AGOR AS’ THEO REM
a
n
M
B
2
A
are product to theThey equalangle. usea isright form hypoten triangle of the gled height right-an squareofofa the The sides hypotenuse. nuse. The two shortest hypotethe thedivides height called side isthe ts in which The longest called the legs.segmen b and 2c. use a and themlegs =h → h =m·n We usually call the hypoten h n
of the two
2
m
M
b
h
Pythagoras’ theorem
A
h
c
M
n
1
B
12 cm
12 cm
2 b2 + c oras’ theorem: a = According to Pythag equal to the sum of the hypotenuse is ons of the the area. of the square ). This means that Apply.. triangle, the projecti of the legs (see figure ❶ 3 In a right-angled 4.5 cm. of the areas of the squares use measure 8 cm and gled.s are similar. two triangle is right-an why these of legs over the hypoten a . Explain if the triangle true 1is Writing only of the legs and the height ship relation This b Calculate the lengths use. hypoten the c and n in this m h, of lengths s 4 Calculate the .2 cm 2, in other words, if they 2 + b 2 31 Pythagorean triple . a = triangle gled c s, c, b, a, satisfy , we say that theright-an If three natural number a right-angled triangle ments of the sides of a = b + c 10 could be the measure 5 cm a few of them: 7 h rean triple. Here are numbers form a Pythago ty of triangle 35, 37 s at 15, 17 the similari 12, n Listening.8,Practise 25 m 3, 4, 13, 84, 85 n.es.41 anayaeducacio 40, 9, 5, 12, 13 16, 63, 65 61 60, 11, 7, 24, 25 kb and ka are too. kc, then triple, a is a Pythagorean ents of the Notice that if c, b, ying each of the compon (the result of multipl For example,846, 8, 10 a Pythagorean triple. triple 3, 4, 5 by 2) is
The sides
mine what of a triangle deter
type it is
, we can find of the sides of a triangle If we know the length it is right-angled: 2 the triangle is right-angled. 2 2 • If a = b + c , . 2 the triangle is an obtuse triangle 2 2 • If a > b + c , . 2 the triangle is an acute triangle 2 2 • If a < b + c ,
out whether or not
Apply... square te the area of the green 1 Writing . Calcula how you following cases. Explain in each of the did it.
b)
a)
45 m2
14 cm2 30 cm2
74
6
60 m2
of four. Now arrange them so that their are six rows of four.
TWO ARE KNOW CALCU LATIN G A SIDE WHEN
• Half joking, half serious!
Here is a cross made of four toothpicks. Can you make a square just by moving one of them?
B’ C’
A’
Watch the video on Target 7.2 and complete
C
A
78
the associated activities at anayaeducacion.es.
Commitment to SDGs
that height of a lamppost Unit 9 at casts a 1.90 m shadow the same time as a bus of shelter with a height 2.40 m casts a 96 cm shadow.
a?
The other leg is 126
c
b?
20 m
a 85
2.6 m d
1m
the wall whose foot is 4 m from wall 3 Reading. A ladder m. How far from the reaches a height of 7.5 it to ladder have to be for would the foot of the m? 8 of reach a height
4
Skills The suggested activities have been specially designed to gradually and continuously develop the four language skills. The main skill worked on in each activity is highlighted:
cm long.
g unknown side of the followin use. where a is the hypoten right-angled triangle, mm a) c = 70 mm; a = 74 cm b) b = 15 cm; a = 25 step with a a ramp to go up a 1 m 2 We want to make is 2.6 m a plank of wood that wheelbarrow. We have step should the ramp start? long. How far from the
The resource bank contains audiovisual material that focuses on Sustainable Development Goals.
b
long, and one of the legs gled triangle is 130 cm The hypotenuse of a right-an other leg. of the is 32 cm. Find the length = 126 2 2 900 – 1024 = 15876 b = 130 – 32 = 16
the 1 Find the length of
obtuse. right-angled, acute or cm a) 15 cm, 10 cm, 11 b) 35 m, 12 m, 37 m dm 21 dm, 30 c) 23 dm, km d) 15 km, 20 km, 25 5 miles e) 17 miles, 10 miles, mm 21 mm, 42 f ) 21 mm, cm 82 cm, g) 18 cm, 80
N
point of the house to the highest of the tablethe two sides, from the edgeand length of (the table must be in a The boy looksis right-an gled,its endweis know in his line of sight side. triangle so that If we know that ). He amoves the ruler to e the length of the third the ruler must be vertical ras’ theorem calculat they we can use Pythago horizontal position, and c, are similar, because d, legs and b n a, know sides s, with from two trianglee gled tenus right-an hypo The the re: Calculating Therefo . are in Thales’ position c b 2 b 2 d+ c=2 a 2 c a2 = b + c → a = the house is equal to of height The c. d, he calculates Knowing a, b and table. the plus the height of the 105 m long. Calculate , the legs are 88 m and In a right-angled triangle length of the hypotenuse. = 137 . 2 Apply.. 2 744 + 11025 = 18 769 a = 88 + 105 = 7 a boat. The ship’s is located 5 m from long. 137 m er boat 2 Ais swimm The hypotenuse level. The mast of the deck is 1 m above sea er's line swimm the In deck. swimmer stands 3 m above the beacon theof the mast and and of theother legthe the topthe sight,when sea of leg . At what height above aligned Calculating one are use lightho c known hypotenuse are level is the beacon? 2 2 2 a2 – c2 → b = a – c 2 2 a2 = b + c → b =
Apply...
s is of the following triangle 2 State whether each
?
B
te the 1 Speaking. Calcula
a d
C
Height rule
• Here you can see twelve counters arranged in three rows
Apply...
b c
c
h
x y – y x
^
al height of a vertic Calculating the its shadow object without using
A
Leg rule
^
^
x y → 10x + y y x → 10y + x
Imagining in space
Unit 10
al object from
height of a vertic Calculating the its shadow
be put in Thales’ position
a
APPL ICATI ONS OF THE
SIMIL ARITY OF TRIAN GLES
8m
SIMIL ARITY BETW EEN
Write any two-digit number and then another number with the same digits swapped round. Subtract one from the other. Can you explain why the difference is always a multiple of 9?
7.5 m
5
practice makes perfect! Use algebra
8.5 m
d 4m te the unknown Listening. Geogebra. Calcula theorem at Pythagoras’ side by applying . n.es ducacio anayae 75
Speaking Listening
Reading Writing
DUAL FOCUS
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MATHEMATICS
2
CE BANK RESOURpanie s this book.
Go to the
Key concepts
SECONDARY EDUCATION
ion.es nayaeducac
at www.a
98-7923I S B N 978-84-6
9
788469
8
879238
In this section of the book, the key information is presented in graphic form, using diagrams and illustrations to help you learn each concept.
SUMA
piezaS
8430902
DUAL FOCUS
Available
…
t r a p is h t in d An
MATHEMATICS
that accom
+
TION SECONDARY EDUCA
du al FOCUS
unit
2
6
Dividing monomials
ALG EBR A
s and the literal parts
We divide the coefficient
Applications of algebra
letter ical language that uses Algebra is a mathemat indeterminate value. ra: algeb Some applications of a · (b + c) = a · b + a ·
ARhombus =
raic A polynomial is an algeb mials that form it. the degrees of the mono omial is the largest of • The degree of a polyn 2x4 – 5x2 + 3x – 8 l omia polyn ree fourth-deg the letters take specific of the monomial when a polynomial is the value • The numerical value of values. = 21 3 · (–2)2 – 2 · (–2) + 5 3x2 – 2x + 5 for x = –2 , but with the has the same monomials which l omia polyn er omial is anoth • The opposite of a polyn ed. chang signs l opposite of a polynomia –7x3 – 3x2 + 5x – 8 7x3 + 3x2 – 5x + 8
2
Monomial by the product raic expression formed A monomial is an algeb or more letters (literal part). one value (coefficient) and mial is the number of
• The degree of a mono
r when
• Two monomials are simila 2y
4x
2
3x + 2x
ials Operations with polynom
2 2·1·2 =8
similar monomials
or subtract similar
Can be calculated Cannot be calculated
2x 3 – 3x + 0x + 6 –x 2 + 5x – 4 + 2 2x 3 – 4x + 5x + 2 2
A –B A–B
2 2x 3 – 3x + 0x + 6 x 2 – 5x + 4 + 2 2x 3 – 2x – 5x + 10
A B A+B
1 x2y 5
10
l
Product of a polynomia and a monomial
Addition and subtraction
l parts. they have the same litera
monomials Adding and subtracting 5a + 2a = 7a
third-degree
literal part when the letters value of the monomial of a monomial is the
take specific values. 2 2ab2 for a = 1 and b =
We can only add monomials.
1+2
2
unit
• The numerical value
–5 ·x ·y 2
part. rs that form the literal
facto
–5 · x · y
coefficient
of a known
Multiplying monomials
cients and literal SIMILARITY We multiply the coeffi . parts of the monomials 2 • 3a · 2a = 3 · 2 · a · a = 6a 3 Similar 2) = 5 · shapes · x · x2 = –15x (–3) • 5x · (–3x Two figures are similar when they only differ in size.
In two similar figures:
Common factor
4 · a + 4 · b = 4 · (a +
F1 F2
(a + b) Thales' ab = a · a + a · b = a · b) a2 +theorem B
A r s
c
b
a
AB = A'B' BC B'C'
a'
α = α'
C
If the lines a, b and c are parallel and intersect two other lines, r and s, then 15 the segments inside are proportional:
α'
a
• The corresponding angles are equal.
2 x 3 – 4x + 5x – 12 x 2 3 4 x 5 – 4x + 5x – x
square of a sum (a + b)2 = a2 + 2ab + b2 e square of a differenc (a – b)2 = a2 – 2ab + b2 squares b2 difference of (a + b) · (a – b) = a2 –
P Q 2·P 3x · P P·Q
2 x 3 – 4x + 5x – 1 3x + 2 2 2x 3 – 8x + 10x – 2 4 – 12x 3 + 15x 2 – 3x 3x 2 3 7x – 2 3x 4 – 10x + 7x +
×
α
×
Notable products
Product of two polynomials
14
mials.
the addition or subtr expression formed by
3x – 2(20 – x) = 40
2a
Polynomials action of several mono
d1 $ d2
Equations
s
Algebraic expression
Double a number
an algebraic fraction
Formulas
sequences
an = (n – 1) · n
c
2 · a ·a = a (2a2) : (6ab) = 2 · 3 · a · b 3b
a monomial
a number
Generalising numerical
properties Expressing arithmetic
quotient could be:
6 $ 3 · x2 · x = 6x 2 18x 3 : 3x = 3 · x2
10 · a2 2 2 10a 2 : 5a = 5 · a2 =
le or
of unknown, variab s to express numbers
of the monomials. The
A'
B'
C'
• The corresponding segments are proportional. b b' Ratio of similarity of F1 andF1Fy2 F2 " a a'
a = b " a = a' a' b' b b' If the ratio of similarity of two shapes is k, then the ratio of their areas is k2.
Similarity of triangles Similar triangles
If the ratio of similarity of two shapes is k, the ratio of their volumes is k3.
C'
B c A
Triangles in Thales’ position
a b
a'
b'
C B'
Two triangles in Thales’ position are similar.
A'
c'
^
3 9
Ratio of similarity Ratio of their volumes
The scale
Calculating the scale
The grid method
Activities The resource bank contains activities for you to practise what you have learnt.
O
Distance in the plan 4 cm = = 4 cm = 1 Real distance 4 m 400 cm 100
E D
B
B
C
a
C'
a'
a a
a
Height rule C A
b C
m a
a = b → b2 = a · m b m
c
h
E' C'
A'
B'
A
F2 B'
C
A
Two triangles are similar if they have two equal angles.
Leg rule
D'
22
^
• they have an equal acute angle, • or their two legs are proportional, • or one leg and the hypotenuse are proportional.
A'
F1
^
C
C'
Similarity between right-angled triangles
Projection method
A
^
Two right-angled triangles are similar if:
C
B
A
^
Their sides are proportional: a = b = c a' b' c'
Building similar figures
Scale: the ratio of similarity between the reproduction and reality.
^
A = A', B = B', C = C '
2 8
a
a' A
Rule on the similarity of triangles
Their corresponding angles are equal: Ratio of similarity Ratio of their areas
B
B'
M
n
m
b
c
h B
a = c → c2 = a · n c n
M
h
A
M
n
B
m = h → h2 = m · n h n 23
7
The
Resource bank www.anayaeducacion.es
This is where you will find resources, techniques and activities to help you consolidate your knowledge. Go to www.anayaeducacion.es, and follow the steps at the beginning of your Spanish book.
un
THE F I N A L CHALL ENGE
Read AND learn To build this mosaic, you need the same number of blue pieces as yellow pieces. In other words, the ratio between the number of diamonds and squares is 1:
it
5
Percentages
No. OF RHOMBU
SES a =1 No. OF SQUARES = a The above statement is easily justified by looking at the following section of the mosaic:
2
calculating percentages
3 1
5
Your book contains references to the online resources you can use to learn more about each topic and to develop your English language skills.
2 5
CANS
AREA YELLOW PART
percentages and proportions
3 4
AREA SQUARE
: for every ... cans of blue paint, you will need ... cans of yellow paint.
percentages and fractions
area of rhombus → 4 area of square → 5
percentages and decimal numbers
INVESTIGATE
8.b and complete
at anayaeducacion.es.
8
calculating the %, given the total and the part
mixtures
meetings and ranges
filling and emptying
Listening. Listen and repeat. The vocabulary is anayaeducacion.es. at interest on loans or taxes on sales, but they haven’t always calculated them the way we do today. The increase in trade during the Renaissa nce led to the standardisation of the modern use of percenta ges.
3 Read the text ‘Percent ages through BANKReading. ucacionNK LANGUAGEanayaed BA .es and answer the questions there. history’ on UAGE BANK GE BANK LANGUAGE NK NK LANG UA LANGUAGE BA NGUAGE BANK LANG BANK LANGUAGE BA NGUAGE 41 NK LA LANGUAGE BA LANGUAGE BANK LABANK LANGUAGE BANK LANGUAGE
All the resources are clearly organised so you can find the ones related to your unit.
res
calculating the total, given the % and the part
1
Resources organised
l Audiovoisuuarces
calculating direct percentagess
2 Discover. People have been using percenta ges since ancient times to calculate the
the associated activities
activities
arithmetic problems
Focus on English
Watch the video on Target
40
Interactive
bank interest
percentage increases and decreases
Look at the two squares cut by Ernesto from a plank of wood. One is twice as long as the other. Knowing that the small square weighs 100 g, we might think that the large square weighs 200 g (double the height, double the weight). However, the large square weighs 400 g, because multiplying the side by two multiplies the area by four. With this in mind, if Paula holds two dice in her hand and the smallest weighs 100 g, how much will the larger die weigh if its edges are twice as long?
by unit
problems with percentages
1
4
However, if you want to draw it and then paint it, you will need more yellow paint than blue paint. • Calculate the ratio between the blue and yellow cans of paint that you need to buy, remembe ring the data from the graph on the right. No. BLUE CANS = AREA BLUE PART = AREA RHOMBUS No. YELLOW • Complete in your notebook
Scan this code to consult the glossary for this unit.
PERCENTAGES
Glossary
QcoRde
un
it
9
Scan this code to consult the glossary for this unit.
Pythagoras’ theorem a
b
Pythagoras’ theorem
c
applications
classifying triangles
a =b +c
finding lengths in polygons
calculating the unknown side of a right-angled triangle
right-angled
calculating the hypotenuse
acute
a?
a = b2 +c2
c
b
obtuse calculating a leg
b? b = a2 – c2
c
a
Focus on English 1
Listening. Listen and repeat. The vocabulary is at anayaeducacion.es.
2 Discover. Pythagoras (5th century BCE) was a Greek mathematician and philosopher who also made some contributions to the study of astronomy. He described how several stars, which he called wandering stars, did not move in the same uniform way as the others. Indeed, the word planet is derived from the Greek for wanderer. This is how the celestial objects that move erratically in the sky came to be called planets.
3
BANK E BANK E G A U NK G A N B A G L E A G U A G U N G A N L
Reading. Read the text ‘An important geometric result’ at anayaeducacion.es and answer the questions.
E BANK
73
1
PYTHAGORAS’ THEOREM
The two shortest sides of a right-angled triangle form a right angle. They are called the legs. The longest side is called the hypotenuse. We usually call the hypotenuse a and the legs b and c. Pythagoras’ theorem
According to Pythagoras’ theorem: a 2 = b 2 + c 2 This means that the area of the square of the hypotenuse is equal to the sum of the areas of the squares of the legs (see figure ❶).
1
This relationship is only true if the triangle is right-angled.
b
Pythagorean triples If three natural numbers, c, b, a, satisfy c 2 + b 2 = a 2, in other words, if they could be the measurements of the sides of a right-angled triangle, we say that the numbers form a Pythagorean triple. Here are a few of them:
3, 4, 5
8, 15, 17
12, 35, 37
5, 12, 13
9, 40, 41
13, 84, 85
7, 24, 25
11, 60, 61
16, 63, 65
a c
a =b +c
Notice that if c, b, a is a Pythagorean triple, then kc, kb and ka are too. For example, 6, 8, 10 (the result of multiplying each of the components of the triple 3, 4, 5 by 2) is a Pythagorean triple.
The sides of a triangle determine what type it is If we know the length of the sides of a triangle, we can find out whether or not it is right-angled: • If a 2 = b 2 + c 2, the triangle is right-angled. • If a 2 > b 2 + c 2, the triangle is an obtuse triangle. • If a 2 < b 2 + c 2, the triangle is an acute triangle.
Apply... 1 Writing. Calculate the area of the green square
in each of the following cases. Explain how you did it. a) b) 45 m2
14 cm2 30 cm2
74
60 m2
2 State whether each of the following triangles is
right-angled, acute or obtuse. a) 15 cm, 10 cm, 11 cm b) 35 m, 12 m, 37 m c) 23 dm, 30 dm, 21 dm d) 15 km, 20 km, 25 km e) 17 miles, 10 miles, 5 miles f ) 21 mm, 42 mm, 21 mm g) 18 cm, 80 cm, 82 cm
2
CALCULATING A SIDE WHEN TWO ARE KNOWN
Unit 9
If we know that a triangle is right-angled, and we know the length of two sides, we can use Pythagoras’ theorem to calculate the length of the third side.
Calculating the hypotenuse from two known legs a = b + c → a = 2
2
2
a?
b2 +c 2
c
b
In a right-angled triangle, the legs are 88 m and 105 m long. Calculate the length of the hypotenuse. a = 88 2 + 105 2 = 7 744 + 11025 = 18 769 = 137 The hypotenuse is 137 m long.
Calculating one leg when the other leg and the hypotenuse are known
b? c
a 2 = b 2 + c 2 → b 2 = a 2 – c 2 → b = a 2 – c 2
a
The hypotenuse of a right-angled triangle is 130 cm long, and one of the legs is 32 cm. Find the length of the other leg. b = 130 2 – 32 2 = 16 900 – 1024 = 15876 = 126 The other leg is 126 cm long.
Apply... 1 Find the length of the unknown side of the following
right-angled triangle, where a is the hypotenuse. a) c = 70 mm; a = 74 mm b) b = 15 cm; a = 25 cm
3 Reading. A ladder whose foot is 4 m from the wall
reaches a height of 7.5 m. How far from the wall would the foot of the ladder have to be for it to reach a height of 8 m?
2 We want to make a ramp to go up a 1 m step with a
4m 2.6 m d
1m
8.5 m
8m
7.5 m
wheelbarrow. We have a plank of wood that is 2.6 m long. How far from the step should the ramp start?
d
Listening. Calculate the unknown side by applying Pythagoras’ theorem at anayaeducacion.es.
4
75
3
APPLICATIONS OF PYTHAGORAS' THEOREM
There are many polygons that have some elements which are sides of a rightangled triangle. This allows us to associate them using Pythagoras’ theorem, and to calculate the length of one of the sides when the other two are known. 1 The diagonals of a rhombus are 10 cm and 24 cm long. Find the perimeter. We start by calculating the length of one side: l
5 cm
12 cm
l = 12 2 + 5 2 = 169 = 13 Each side is 13 cm long. The perimeter is: P = 4 · 13 = 52 cm
2 Find the area of an isosceles trapezium with bases of 30 cm and 48 cm, and an oblique side of 41 cm. Remember that the area of a trapezium is: A = (b + b' )· h 2 b = 30 cm We start by calculating the height, h. 41 cm
h
9 cm b' = 48 cm
The short side of the green triangle is (48 – 30) : 2 = 9 cm. h = 41 2 – 9 2 = 1600 = 40 The height of the trapezium is 40 cm. A = (30 + 48) · 40 = 1 560 cm2 2
3 Calculate the area of an equilateral triangle with sides that are 8 cm long. We begin by calculating the height: h
8 cm
4 cm
h = 8 2 – 4 2 = 48 ≈ 6.9 The height is approximately 6.9 cm. The area is: A = 8 · 6. 9 = 27.6 cm2 2
4 Calculate the radius of the circumscribed circumference of this square with sides of 10 cm. 10 cm r
r
The diagonals of the square are perpendicular. Therefore, the coloured triangle is right-angled and has legs of equal length. r 2 + r 2 = 102 → 2r 2 = 100 → r 2 = 50 r = 50 = 7.07 cm
76
Unit 9 5 Calculate the area and perimeter of a regular pentagon with an apothem of 16.2 cm and a radius of 20 cm. l/2 16.2 cm
20 cm
First we calculate the length of one side: l = 20 2 – 16. 2 2 = 137. 56 ≈ 11.7 2 The side of the pentagon is: l = 11.7 · 2 = 23.4 cm Therefore, its perimeter is: P = 23.4 · 5 = 117 cm
Finally, we calculate the area: Perimeter · apothem 117 · 16. 2 = 947.7 cm2 = 2 2
A=
6 Find the length of a circumference on which a 6.6 cm chord has been drawn 5.6 cm from the centre. Calculate the area of the corresponding circle. We start by calculating the radius. The length of the shortest side of the coloured triangle is k = 6.6 : 2 = 3.3 cm. Therefore: r = 3. 3 2 + 5. 6 2 = 42. 25 = 6.5
5.6 cm r
k
6.6 cm
The radius is 6.5 cm. P = 2πr = 2 · 3.14 · 6.5 ≈ 40.8 cm A = πr 2 = 3.14 · 6.52 ≈ 132.7 cm2
Apply... 1 Find the area of an equilateral triangle with a
perimeter of 54 cm.
2 Find the area and perimeter of an isosceles trapezium
with bases of 3.2 m and 6.4 m and a height of 6.3 m.
3 Writing. Read and solve the following problem
explaining the method you used. Calculate the area of a regular hexagon with sides of 18 cm. (Remember that in a regular hexagon the sides and the radius are the same length).
4 A chord of 13 m is drawn on a circumference with
a radius of 9.7 m. How far is the chord from the centre of the circumference?
5 A regular pentagon is inside a circle with a radius of
1 m. The perimeter is 5.85 m. Calculate the area.
6 Speaking. Discuss how to solve the following
problem as a group and then solve it. Find the length of the diagonal of a cuboid whose dimensions are 8 dm, 6 dm and 14 dm.
77
THE F I N A L CHALLENGE INVESTIGATE How to mark out a beach volleyball court We want to mark out a beach volleyball court. How do we draw the lines? The best way is with a rope pulled tight. How do you get the right angle in the corners? Take a rope and mark twelve identical sections by making knots. With three stakes, tighten the rope to form a triangle with 3, 4 and 5 knots on the sides. It is a right-angled triangle. The right angle is at the vertex where the sides with 3 and 4 knots meet. A bit of history Over 3 000 years ago, the Egyptians used this method to draw right angles. The river Nile would burst its banks and flood farmland every year, and the borders between the flooded fields needed to be restored. The land surveyors who were responsible for marking the borders again used the method described above.
practice makes perfect! Use algebra Write any two-digit number and then another number with the same digits swapped round. Subtract one from the other. Can you explain why the difference is always a multiple of 9?
x y → 10x + y y x → 10y + x
x
?
y – y x
Imagining in space • Here you can see twelve counters arranged in three rows
of four. Now arrange them so that their are six rows of four.
• Half joking, half serious!
Here is a cross made of four toothpicks. Can you make a square just by moving one of them?
Watch the video on Target 7.2 and complete 78
the associated activities at anayaeducacion.es.
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9
PYTHAGORAS’ THEOREM
Pythagoras’ theorem a2
c2
Pythagoras’ theorem states that: a² = b² + c²
c a
b
This relationship is only true if the triangle is right-angled. a, b and c form a Pythagorean triple.
b2
Classification of triangles using Pythagoras’ theorem
b ac
a b c
b ac
right-angled a2 = b2 + c2
obtuse a2 > b2 + c2
acute a2 < b2 + c2
Calculating one side of a right-angled triangle using Pythagoras’ theorem Calculating the hypotenuse from two known legs
Calculating one leg when the other leg and the hypotenuse are known
b? a?
c
c
a
b a = b2 +c 2
20
b = a2 – c 2
Calculating lengths in polygons using Pythagoras’ theorem Diagonal of a rectangle
Side of a rhombus l
5 cm
89 cm
b
12 cm
80 cm
b = 89 2 – 80 2 = 39 cm
l = 12 2 + 5 2 = 13 cm
Height of a triangle
Height of a trapezium b = 30 cm h
41 cm
8 cm
h
9 cm b' = 48 cm
h = 41 2 – 9 2 = 40 cm
Side of a regular pentagon
4 cm
h = 8 2 – 4 2 c≈ 6.9 cm
Radius of a circumference
l/2 16.2 cm
20 cm
5.6 cm r
l = 20 2 – 16.2 2 ≈ 11.7 2 l = 11.7 · 2 = 23.4 cm
k
6.6 cm
k = 6.6 : 2 = 3.3 cm r = 5.6 2+ 3.32 = 6.5 cm
21