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s c i t a m e h t a M Albero u l e t z Ga nez, I. é m i J s ra a Caña r e J. Cole l o C R.

Building

Blocks

this is

your book

Reading and listening

CH UNIT

F EA THE OPENING PAGES O

We read and listen a brief historical introduction of the contents your are going to learn in the unit.

You can do these motivating activities to activate your previous knowledge.

Pythagorean theore

Pythagorean theorem is right-angled: When the blue triangle A=B+C

PYTHAGOREAN THEOREM

square is equal to the The area of the large two small ones. sum of the areas of the

Reading and listenin

3 What

Babylonians Egyptians and ical concept. As you is an important geometr of the sides of any the squares The Pythagorean theorem relationship between know, it describes the knew right-angled triangle. s and the Babylonians ago, both the Egyptian them to construct used they More than 3 000 years sides were right-angled and , used triangles whose that certain triangles Egyptians, for example sacred) to divide fields and right angles. The ed 5 (which they consider and 4 3, d measure build pyramids.

cm2? 2 = 9 cm and C = 16 is the area of A if B

icular to Euclid’s proof drew a line perpend rean theorem, Euclid large square To prove the Pythago line), dividing the the triangle (the green and C = A2. the longer side of = A1 He showed that: B into two rectangles. A1 A2

B C

Egypt and Pythagoras th BCE) travelled to ras (6 century es. His great In his youth, Pythago about these properti g the undoubtedly learnt Babylon, where he general theorem describin gled come up with the right-an achievement was to the sides of any squares drawn on relationship between his name. this theorem bears why is That triangle.

and answer. 4 Look at the figure the grid as Taking a square from A1 a unit: A2 squares does a) How many unit 9 B, contain? the small square, do you What ? A 1 rectangle 16 And B 15 notice? number of unit 20 b) Check that the in as same the squares in C is A2. C . c) Describe this property it confirm? d) What theorem does State the theorem. cm. Write 5 cm, 12 cm and 13 of sides with gled triangle the lengths of the sides. 5 Draw a right-an g the relationship between an equality expressin

ts Euclid, Euclid's Elemen the theorem, but Pythagoras who proved Elements around the year However, it was not ria wrote d and Euclid of Alexand he collected, organise g it two centuries later. of 13 books in which ge of his time, providin 300 BCE. It is a set the mathematical knowleddemonstrated what we now expanded upon all I he Book In . structure with a solid logical ean theorem. Pythagor the as know verb ‘come up with’. to define the phrasal verb in 1 Use context clues k using this phrasal sentence in your noteboo Then, write a new . the a different situation write sentences about ending and past continuous, with Pythagoras and 2 Using past simple starting rean theorem history of the Pythago with Euclid.

LANG

178

CONTENT DEVELOPMENT AND ACTIVITIES 3 APPLICATIONS OF THE PYTHAGOREAN THEOREM

PYTHAGOREAN THEOREM

Consolidating

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. In general, we say that a is the hypotenuse and b and c are the legs. a

b c

This observation was made by the Chinese 400 years before Pythagoras was born. anayaeducacion.es GeoGebra. Graphical demonstration of the Pythagorean theorem.

Look at this demonstration:

If we compare both figures, it is clear that a 2 = b 2 + c 2.

a c

What are the areas of the unknown squares in the following shapes?

102 m2

2 = reduce 1.2 m the length of the Looking at the blue triangle: AC = 1.2 2•+If…we … m the size of the right angle making it acute, Therefore: 2 = … mside decreases, as does the area of the corresponding B square. Looking at the red triangle: d = … 2 + 4.8opposite 1.6 m A 2 2 2 2 2 2 The diagonal of the cuboid measures 5.2 m.Figure C 8 4.5 is less than 3 + 4 8 In general: a <Db + c 4.8 m ideas Note that you can calculate it directly: idating Consol

d = 1.2 2 + 1.6 2 + 4.8 2 = f = 5.2 m

In general, in a cuboid with dimensions a Ò b Ò c the diagonal is: a) b) d = a 2 + b 2 + c 2 A1

57 cm2

87 m2

B 1.6 m

1.2 m A 3 dm2

13 Calculate the height, h, of a regular pyramid that has a square base with 30 cm

sides, and a lateral face with an area of 255 m2. cm2 the height, h, of the lateral face. First, we 57 find 31 m2 A 255 = 30 · a 8 510 = 30a 8 a = … m A1 = 57 + 57 = … cm2 2 A2 = 87 + … = … m2 The height of the lateral face is the hypotenuse of the triangle shown on the right:

8, 15, 17

12, 35, 37

5, 12, 13

9, 40, 41

13, 84, 85

7, 24, 25

3, 4, 5

11, 60, 61

16, 63, 65

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.

70 2 + 240 2 = 4 900 + 57 600 = 62 500

Let’s practise! 245 2 = 60 025

of 54 cm. 15 2 + 36 2 = 225 + 1296 = f

d) 15 km, 20 km, 25 km a

Examples and solved problems. To put into practice the most important methods.

Find the diagonal of a square with a perimeter of 28 dam.

The parallel sides of a right trapezium are 13 dm and 19 dm long, and the oblique side is 10 dm. Calculate the height.

Calculate the identical sides of an isosceles triangle, knowing that the non-identical side is 5 m long, and the corresponding height is 6 m.

Calculate the length of the side of a rhombus with diagonals of 1 dm and 2.4 dm.

Find the height of an equilateral triangle with 40 cm sides. Round to the nearest millimetre.

Find the apothem of a regular hexagon with 20 cm sides. Remember that in a regular hexagon the side and the radius have the same length.

A regular pentagon with 11.7 cm sides is inscribed in a circumference with a radius of 10 cm. Calculate the apothem.

A straight line passes 10 cm from the centre of a circumference that has a radius of 15 cm. Find the length of the resulting chord, rounding your answer to the tenths.

How far from the centre of a circumference with an 8 cm radius must a line pass so that the chord measures 8 cm?

Calculate the diagonal of a cube with 20 cm sides. Round to the nearest millimetre.

f ) 21 mm, 42 mm, 21 mm g) 18 cm, 80 cm 82 cm

15 m

a = … dm2 A3 = 14h – …

Calculate the unknown side in each right-angled triangle: a)

b) 65 mm

15 m 20 m

4 A 13-metre chord is drawn on a circumference with

a radius of 9.7 m. How far is the line from its centre? 4 8 392 … 152 + 362 8 The triangle is… 2 = 1521 2 Find the and perimeter of an isosceles trapezium 5 A regular pentagon is inscribed in a circumference 39area with 3.2 m and 6.4 m bases, and 6.3 m height. with a radius of 1 m. Its perimeter is 5.85 m. Calculate c) 18 m, 80 m, 83 m the area. 3 Calculate the area of a regular hexagon with 18 cm + 6a400 = 6 724 18 2 + 80 2 = that 324 in length of the sides. (Remember regular hexagon 2 8the 4 8 the 832side … 182 +680Find The triangle is…diagonal of a cuboid with sides 2=f of 8 dm, 6 dm and 14 dm. and the83 radius have the same length.)

186

Consolidating Ideas. Exercises to co mplete and consolidat e the theory that the teacher has explained to you.

e) 17 miles, 10 miles, 5 miles

16 mm

Calculate the unknown side of each triangle and round it off to the tenths. a)

16 m 12 cm

anayaeducacion.es GeoGebra. Calculating areas applying the Pythagorean theorem. 180

12 dm

b) 35 m, 12 m, 37 m

14 dm2

Calculate the perimeter of a rectangle with a diagonal of 5.8 cm, and one side that measures 4 cm.

Say whether each of the following triangles is right-angled, acute-angled or obtuse-angled:

c) 23 dm, 30 dm, 21 dm

4 8 2452 < 702 + 2402 8 The triangle is…

15area dm,of36 39 dm triangle with a perimeter 1 Findb)the andm, equilateral

4 cm

a) 15 cm, 10 cm, 11 cm A

17 cm

21 dm

Calculate the area of the following squares:

2 – …2 = 2 Copy and complete to find each 30 m h = out …whether fof=the 8 mfollowing triangles is right-angled, acute or obtuse: 30 m Thea)pyramid 8m high. 70 cm, is 240 cm, 245 cm

ã 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:

F ocus on English

30 cm2

1 Copy these shapes in your notebook. Draw the square that is missing in each one and say what4.8the m area is. C

S1 386 dm2

acute triangle

If wecm expand the right angle making it obtuse, the length of the opposite side r = f =• 7.07 increases, as does the area of the corresponding square. Therefore:

60 m2

14 cm2

10 cm

1 cm

45 m2

Find the perimeter of the following shape:

Calculate the area of the green square in each of the following cases: a)

D 2 is greater 12 Find the diagonal of a cuboid with dimensions of B 1.28m,61.6 m and 4.8 thanm.32 + 42 8 In general: a 2 >C b 2 + c 2 Figure

As both triangles are right-angled, in both cases the area of the biggest square is equal to the sum of the areas of the smaller squares. Therefore: S2 = 386 dm2 – 47 dm2 = 339 dm2 S1 = 183 m2 + 102 m2 = 285 m2

satisfy: to fulfill a condition.

23 cm

• We already the relationship between the areas of the squares built on the 10 cm. Remember that the diagonals of the square are know perpendicular. r r sidesand of ahas right-angled triangle: Therefore, the coloured triangle is right-angled legs of equal length. 2 is equal to 32 + 42 8 In general: a 2 = b 2 + c 2 =… A 88r 25= … r 2 + r 2 = …2 8 2r 2Figure

Problem solved

47 dm2

18 cm

r O

… ≈ 643.7 •A Ifcircle a2 < = b 2πr + c 2=, … the ·triangle is an cm acute triangle. right-angled triangle obtuse triangle 11 Calculate the radius of the circumscribed circumference of this square with sides of

Unit 9

Pythagorean theorem

Therefore, the triangle PTO has a right angle at T :

unit

from this resources to choose Remember io. for your portfol

• If a 2 > b 2 + c 2, the 2 is an 2 2 triangle 2 2 obtuse triangle.r = OP – PT = … – … = …

183 m2

The icons included with some activities indicate the keys to the project.

10Remember The distance of a point P to the centre O of aAcircumference is OP B= 23 cm. We If draw we know the sidesfrom of a P triangle, a tangent to thewe circumference. The PT tangent segment is 18 cm. can findthe outarea whether it iscircle. right-angled: Find of the

The two identical squares have b + c as sides.

Look at the diagrams and analyse the explanation below:

• The If a 2tangent = b 2 + c 2line , the is is triangle perpendicular to the radius. right-angled.

According to the Pythagorean theorem: a 2 = b 2 + c 2 This means that the area of the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs. This relationship is only true if the triangle is right-angled.

a2 = b2 + c2 Interesting fact!

Unit 9

ã The sides of a triangle determine its type

ideas

12 cm

divided Each unit is phs and into epigra hs. subepigrap portant The most im in bold. re a contents

Focus on English. Do you think Mathematics and English haveanything PROBLEMS EXERCISES AND in common? Discover how language and mathematics are linked so you can learn both: Mathematics and English.

17 m

32 mm

28 mm

The audios of each unit’s content are available at www.anayaeducacion.es

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Find the diagonal of a cuboid with sides of 3 cm, 4 cm and 12 cm.

A 24 m tangent segment is drawn from an external point P to a circumference with a 10 m radius. How far is P from the centre of the circumference?

181

187

Let’s practise! These are exercises to apply the theory you have learnt.

KEYS

PROJECT

SDG SDG Commitment Discover the Sustainable Development Goals and be an active part of our commitment to make a more equal and liveable world.

Developing thinking Work on strategies for thinking: reflect on the content you are learning, generate ideas, organise them, debate them, explain them…

Cooperative learning Get involved in your learning and participate in the group’s learning; you will find that cooperating improves performance and harmony in the class.

Emotional education Get to know yourself; identify the situations that bring up complicated emotions and manage them with constructive, self-affirming experiences.

EXERCISES AND PROBLEMS 3 APPLICATIONS OF THE PYTHAGOREAN THEOREM

from this resources to choose Remember io. for your portfol

unit

Unit 9

PROBLEMS EXERCISES PROBLEMS S AND EXERCISEAND

idating ideas

Consol

18 cm

23 cm

geometrical shapes. Round to one decimal place. Calculate the area of the green square in each of a)the following cases: b) a) b)

13 Calculate the height, h, of a regular pyramid that has a square base with 30 cm

sides, and a lateral face with an area of 255 m2. First, we find the height, h, of the lateral face. 255 = 30 · a 8 510 = 30a 8 a = … m 2 The height of the lateral face is the hypotenuse of the triangle shown on the right: h = …2 – …2 = f = 8 m

Glossary A

associative property billón

The pyramid is 8 m high.

additive system

5.6are dm grouped. g) 18not cm,depend 80 cm on 82 how cm the sumands c) A 1 followed by 12 zeros d) (One million millions). Calculate the unknown side in each right-angled To join several quantities (addends) into one. triangle:

commutative property

Let’s practise!

decimal numeral system

1 Find the area of an equilateral triangle with a perimeter

4 A 13-metre chord is drawn on a circumference with

2 Find the area and perimeter of an isosceles trapezium

5 A regular pentagon is inscribed in a circumference

with 3.2 m and 6.4 m bases, and 6.3 m height.

3 Calculate the area of a regular hexagon with 18 cm

sides. (Remember that in a regular hexagon the side and the radius have the same length.)

a radius of 9.7 m. How far is the line from its centre? distributive property 5

186

6 Find the length of the diagonal of a cuboid with sides integer division

of 8 dm, 6 dm and 14 dm.

million

multiplication anayaeducacion.es GeoGebra. Calculating areas applying the Pythagorean theorem. natural numbers

10 m

8 dm

3 dm

d 10

Find the height of an equilateral triangle with 40 cm sides. Round to the nearest millimetre.

13d)

Find the apothem of a regular hexagon with 20 cm 5m sides. Remember5 mthat in a regular hexagon the side and the radius have the same length.

15 23

Problem solving

Is this triangle right-angled?

Calculate the length 4 dm of the side of a rhombus with diagonalsmof 1 dm and 2.4 dm.

Problem solved 28

20 cm

13 cm

A 14.5 m-high electricity pole breaks at the base and falls against a building located 10 m away from it. At what height does the pole hit the building?

5 cm

Solution: We first calculate the unknown side CB . A To express a number as a product of its divisors. 20 13 of athe common divisors of two or more numbers. The greatest

divisor (GCD)

A straight line passes 10 cm from the lowest centre of common a circumference that has a radius of 15 cm. multiple Find the(LCM) length of the resulting chord, rounding your answer to multiple Problem the tenths. solved

C of the common multiples B of two or more numbers. The lowest 5 M x 2

During a carnival in my town, we hang

a = 13 – 5 = 144 8 a = 144 8 a = 12 a 1 m-high piñata in the middle of a 34 m-long rope, Number containing another number an exact number of times. which is tied to two 12 m-high poles that are 30 m apart. How far from the ground is the piñata?

2 = 202 – a 2 = 256 8 x = 256 8 x = 16 xNumber that is only divisible by itself and the unit.

CB = 5 + x = 21 cm 20

17 Calculate the diagonal of a cube with 20 cm sides. Solution: 4. INTEGERS Calculate side that of each A propertythe of unknown multiplication saystriangle that theand product of the not 2 +multiplication 2 =millimetre. x 2 =to42the8nearest 2xdoes 16 xRound absolute value The natural number21we get from removing its sign. round it off to the2 remove tenths. the brackets. change if we km 2 integer 8 the x = diagonal 8 ≈ 2.83of a cuboid with sidesof ofan 3 cm, a) The distribution b) of a whole among c) several, equal parts.18 x = 8Find 13 2 + 20 2 = 569 4 8 212 < 132 + 202 4 cm and 12 2cm. opposite of an integer Another A = x $ x = x = 8 = 4 mm2 21 2 = integer 441 with the same absolute value, but with the opposite sign. Division where the remainder is zero. 2 2 2 32 mm x set The set of all positive natural numbers, zero, and the negatives of the natural 16 m 17 m 19 A 24 m tangent segment is drawn from anZ external Division where the remainder is not zero. It is an acute triangle. P = x + x + 4 = 2 · 2.83 + 4 = 9.66 mm numbers. point P to a circumference with a 10 m radius. How 30 m A 1 followed by 6 zeros. 12 cm far is P from the centre of the circumference?

division with a radius of 1 m. Its perimeter is 5.85 m. Calculate the area. exact division

We learn the relevant terms that are underlined in the units with a clear definition.

39 m

16 m

Numeral system where adding symbols adds their represented amount. prime number b) 65 mm 22 cm x 9.6property cm Calculate theof area and perimeter mmsays that 16the result How far from thedoes centre of a circumference with A of addtion and multiplication 16 that the sum 15 m of the that coincides with not changexif the order of the sumands changes. an triangle 8 cm radius must a line pass so that the chord 10 m half of a square whose diagonal Positional numeral system with ten symbols or figures (0,measures 1, 2, 3, 4, 8 cm? 5, 6, 7, 8 and 9). x 20 m measures 4 mm. 4 mm This is the numeral system we currently use.

12 cm

of 54 cm.

25 mm

addition

The exercises are divided into topics. Each one is also marked with its degree of difficulty, from one to three.

9 mm

13 m

inscribed 5Amregular pentagon with 11.7 cm sides is factorise in a circumference with a radius of 10 cm. Calculate greatest common the5 result apothem. f ) 21Amm, 42 mm, 21 mm and multiplication that says that the m property of addition of the sum does

15 m

30 m

12 dm

Areas and perimeters using the 3 Pythagorean Say whether theorem each of the following triangles is 21 right-angled, Find the area and perimeter of these shapes. To do acute-angled or obtuse-angled: so, will 10 firstcm, have a) you 15 cm, 11to cmcalculate the unknown length of one of their elements. If they are not exact, find b) 35tom, 12decimal m, 37 m them one place. c) 23 dm, 30 dm, 21 dm 2.4 dm b) a) d) 15 km, 20 km, 25 km

25 mm 1. NATURAL NUMBERS e) 17 miles, 10 miles, 5 miles

h h

30 m

20 d

m The20parallel sides of a right trapezium are 13 dm and 19 dm long, and the oblique side is 10 dm. Calculate the height.

3m Calculate the identical sides of an isosceles triangle, knowing that the non-identical side is 5 m long, and c) the corresponding height is 6 m.

21 dm

1.2 m 1.6 m

Find the diagonal of a square with a perimeter of 28 dam.

1.6 m

4.8 m

C B

d = a 2 + b 2 + c 2

x 36 m

3 dm

mm 20

28 mm

d = 1.2 2 + 1.6 2 + 4.8 2 = f = 5.2 m

4 cm

20 cm

4.8 m

In general, in a cuboid with dimensions a Ò b Ò c the diagonal is:

Glossary.

17 cm

1.2 m

Looking at the red triangle: d = … 2 + 4.8 2 = … m The diagonal of the cuboid measures 5.2 m. Note that you can calculate it directly:

Classify the following triangle as either a rightangled, acute or obtuse triangle. To do this, calculate some of its elements.

Looking at the blue triangle: AC = 1.2 2 + … 2 = … m

5 cm

r = f = 7.07 cm

12 Find the diagonal of a cuboid with dimensions of 1.2 m, 1.6 m and 4.8 m.

Calculate the area of the following squares: 12 m

Calculate the measurements needed to classify the following triangle according to its angles.

10 m

diagonal of 5.8 cm, and one side that measures 4 cm.

c) 2

32 cm

12 cm the perimeter of 7 13 cm Calculate a rectangle with a 20 cm

0 cm

20 cm

15 m 30 cm2

24 cm

10 cm. Remember that the diagonals of the square are perpendicular. Therefore, the coloured triangle is right-angled and has legs of equal length. r 2 + r 2 = …2 8 2r 2 = … 8 r 2 = …

60 m2

14 cm2

10 cm

11 Calculate the radius of the circumscribed circumference of this square with sides of

45 m2

26 cm

r 2 = OP 2 – PT 2 = …2 – …2 = … Acircle = πr 2 = … · … ≈ 643.7 cm2

Calculate the area and the perimeter of these shapes. Note that in the first two the perimeter is the inner and outer periphery.

mm 15

25 m

draw a tangent from P to the circumference. The PT tangent segment is 18 cm. Find the area of the circle. The tangent line is perpendicular to the radius. Therefore, the triangle PTO has a right angle at T :

Find thethe area and perimeter these shapes. Find perimeter of the of following shape:To do so, you will first have to calculate the1unknown length cm of one of their elements. If they are not exact, find them to one decimal place.

12 m

10 The distance of a point P to the centre O of a circumference is OP = 23 cm. We

Exercises and problems. For you to apply all the contents that you learnt throughout the unit.

22 6

20 Calculate the length of x for each of the following Pythagorean theorem

A repeated addition of the same value.

188

Numbers that can be used to count items.

189

187

numeral system

Set of symbols and rules used to represent numbers.

5. DECIMALS

positional numeral system

Numeral system where symbols have different values depending on their level.

subtraction

decimal number

To remove an amount (subtrahend) from another (minuend) to find out the difference between the two.

The result of a non-exact quotient. It has an integer portion and a decimal portion, separated by a decimal point.

hundredth

The result of dividing one tenth into ten equal parts.

number line

A one-dimensional line that contains all the real numbers.

tenth

The result of dividing one into ten equal parts.

thousandth

The result of dividing one hundredth into ten equal parts.

unit

The element used to build all natural numbers, represented by the number 1.

trillón

A 1 followed by 18 zeros (One million billones).

2. POWERS AND ROOTS power

A shortened form of writing a product of equal factors.

power of base 10

The unit followed by as many zeros as figures marked in the exponent.

product of powers with the same base

To multiply two powers with the same base, we keep the same base and add the exponents together.

quotient of powers with the same base

To divide two powers with the same base, we keep the same base and subtract the exponents.

3. DIVISIBILITY

6. THE METRIC DECIMAL SYSTEM angstrom

A unit used to measure atomic distances.

astronomical unit

The average distance from the Earth to the Sun. It is used to measure the distance between planets.

gram

The main unit for measuring masses.

light year

The distance light travels in one year. It is used to measure the distance between galaxies.

divisible

Number that when divided by another gives an exact result.

litre

The main unit for measuring capacities.

composite number

A number that can be factorised into simpler factors.

magnitude

Quality and property of objects that can be measured and quantified numerically.

divisor

Number that is contained in another number an exact number of times.

metric decimal system

The set of units of measurement for basic magnitudes.

factor

Each of the quantities that can be multiplied to form a product.

micrometre

One thousandth of a millimetre.

290

291

MATHS WORKSHOP Unit 9

MATHS WORKSHO READ AND LEARN

PRACTICE MAKES PERFECT!

Pythagoras

Use algebra

Pythagoras (6th century BCE) was known as a mathematician and philosopher. However, his contributions to astronomy are not very well known.

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?

— Moreover, he was one of the first people to realise that some stars, which he called wandering stars, did not have the same regular movement as the other stars. In fact, the word wandering in Greek is pronounced planet. This is why planets were called celestial objects that wander in the sky. We now know that planets are not related to stars. This is why they appear to wander.

INVESTIGATE

• Here is a cross made of four toothpicks.

SELF-ASSESSMENT

b) x

16 m

It is a right-angled triangle. The right angle is at the vertex where the sides with 3 and 4 knots come together.

72 mm

21 mm

shapes: a)

With three stakes, tighten the rope to form a triangle with 3, 4 and 5 knots on the sides.

23 .4

30 m

40 dm

24 cm

30 m

Over 3 000 years ago, the Egyptians used this method to draw right angles.

d) m 31

Every year, after the River Nile’s floods, the borders between the flooded fields needed to be restored.

4 A town square has the shape and the dimensions

shown in the picture. All the angles marked in red are 45°. Calculate the area and perimeter of the square. 12 m

Commitment

4 cm

Remember that you can find academic and professional guidance related to this content at anayaeducacion.es.

40 cm 25 cm

a 40 cm

8.66 m

192

26 cm

26 m

4c m

A bit of history

The land surveyors who were responsible for marking the land borders again used the method shown on the left.

Trust in your skills and knowledge, develop creativity, adapt to changing situations and have a proactive and responsible attitude.

2 Calculate the unknown segment in each of these

Take a rope and mark twelve identical sections by making knots.

Enterprising culture

3 Calculate the area and perimeter of these shapes: 8m

How do you get the right angle in the corners?

Can you make a square just by moving one of the toothpicks?

10 mm

How do we draw the lines? The best way is with a rope pulled tight.

anayaeducacion.es Answer key and interactive self-assessment.

acute-angled or obtuse-angled: a) 20 cm, 24 cm, 30 cm b) 5 m, 6 m, 10 m c) 10 mm, 24 mm, 26 mm d) 7 dm, 7 dm, 7 dm

We want to mark out a beach volleyball court.

Here, you will find readings, activities, advice, information...

Half joking, half serious!

• Here you can see twelve counters arranged in three rows of four. Now arrange them so that there are six rows of four.

1 Classify the following triangles as right-angled,

How to mark out a beach volleyball court

x y – y x

Imagining in space

— He was also the first person to find out that the Moon’s orbit is not in Earth’s equatorial plane, but inclined to it by a certain angle.

Practice makes perfect! In this section you will have to solve many different types of problems.

— He was the first Greek to recognise that the star that we can seen in the morning and at dusk was the same star. We now know that this star is the planet Venus.

x y 8 10x + y y x 8 10y + x

4m 10 m

Watch the video for target 7.2. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.

26 m

ment. Self-assess ies, ese activit th By doing r u o y heck you can c how ding and n ta unders t. rn a le e v ha much you

193

Academic and professional

ICT

orientation

Assessment

Linguistic Plan

Learn how to obtain information, select it and apply it; to plan, manage and work on projects; to collaborate online in an ethical and safe manner.

Evaluate your personal skills, discover and awaken your calling, train yourself to make decisions and learn to choose between different options.

Discover different strategies to analyse what you have learnt and how you learnt it; train yourself to take responsibility or overcome difficulties.

Use your communication skills in the different types of text that you will see. Language is always present, communicate!

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BANK

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A digital version of your book to be used online or offline. It offers access to your digital resources which are grouped by type or linked to unit content.

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Resources related to THE PROJECT KEYS

SDG

SDG Commitment with short videos that will help you understand the targets for reaching the Sustainable Development Goals worked on in this project. Linguistic Plan with infographics that will give you models to work with the four linguistic skills, using different text types (descriptive, narrative, explanatory, etc.).

Cooperative learning Preparing for the task In small groups, of four or five members: 1 All the members of the team will review how the assigned task can be accomplished. 2 To do this, the steps can be shared out to each team member who then in turn will explain how each part of the process can be done to the others. The others listen and participate if they think they can contribute something.

Authorship / adaptation : Variant of the Educational Innovation Laboratory of the colegio Ártica - David and Roger Johnson.

Developing thinking with explanations are included on how to apply the different thinking techniques proposed in the project.

3 Once everyone is in agreement on how to do each part, you will all complete the tasks and, finally, verify, among everyone, that you have solved it correctly.

Cooperative learning which includes the descriptions of the cooperative learning techniques proposed in the project. Thinking techniques

Emotional education with resources to help you overcome any worries that may arise in different situations at school (beginning of the school year, taking a test, etc.).

Logic Wheel This thinking technique will help you to establish phases when analysing specific content that you have to study.

Identify What is it? What is it like? Are there different types?

By following a logical sequence (the logic wheel), and by asking yourself a series of questions in each phase, you can:

• Identify content by asking yourself: What is it? What is it like? Are there different types? • Compare the content by formulating questions such as: In what way is it similar to ...? In what way is it different from ...? • Establish cause-effect relationships by asking yourself questions such as: Why? What impact does it have ...? • Argue, assess and ask yourself questions such as: What conclusions can be drawn after the analysis? What can be assessed or scored about it? Doing a data dump of these questions into a graphic organiser will help you.

Compare

Argue, assess What can we conclude?

Logic wheel

In what way is it similar to ...? In what way is it different from ...?

3 Establish cause-effect relationships Why? What impact does it have ...?

Authorship: Hernández, P., and García, L. A.; adapted by Escamilla, A.

ICT resources to help you use information and communication technology in a healthy, correct and safe way.

Academic and professional orientation with information on different professions linked to the subject content.

0:40/1:33

Assessment which includes resources for your portfolio, as well as rubrics and targets that will help with your self-assessment.

resources SUBJECT KEY CONCEPTS Tutorials Key concepts Activities with GeoGebra Learn by playing Glossary Self-assessments Language Bank

Resources classified by unit

All the resources are classified by unit so that you can find them more easily.

course contents NATURAL NUMBERS AND INTEGERS

Page 8

1. The set of natural numbers.............................................. 10 2. The relationship of divisibility......................................... 12 3. Prime and composite numbers...................................... 15 4. Lowest common multiple of two or more numbers 16 5. Greatest common divisor of two or more numbers 17 6. The Z set of integers.......................................................... 18 7. Operating with integers.................................................... 19 8. Powers of integers............................................................... 23 9. Roots of integers.................................................................. 25 Exercises and problems......................................................... 26 Maths workshop........................................................................ 30 Self-assessment........................................................................ 31

DECIMAL NUMBERS AND FRACTIONS

Page 32

1. Decimal numbers................................................................. 2. Operating with decimal numbers.................................. 3. Decimal and sexagesimal numbers............................... 4. The square root of a decimal number......................... 5. Fractions.................................................................................. 6. Fractions and decimal numbers..................................... Exercises and problems......................................................... Maths workshop........................................................................ Self-assessment........................................................................

OPERATING WITH FRACTIONS

34 38 42

PROPORTIONALITY

PERCENTAGES

Page 94

1. Percentages. Concept........................................................ 96 2. Problems with percentages............................................. 99 3. Bank interest.......................................................................... 103 4. Other arithmetic problems............................................... 104 Exercises and problems......................................................... 107 Maths workshop........................................................................ 110 Self-assessment........................................................................ 111

ALGEBRA

Page 112

1. Why do we use algebra?................................................... 114 2. Algebraic expressions........................................................ 116 3. Polynomials............................................................................ 119 4. Notable products................................................................. 122 Exercises and problems......................................................... 125 Maths workshop........................................................................ 130 Self-assessment........................................................................ 131

43 44 46 48 54 55

Page 56

1. Adding and subtracting fractions.................................. 58 2. Multiplying and dividing fractions................................. 60 3. Problems with fractions.................................................... 62 4. Powers and fractions.......................................................... 66 Exercises and problems......................................................... 70 Maths workshop........................................................................ 74 Self-assessment........................................................................ 75

Page 76

1. Ratios and proportions...................................................... 78 2. Directly proportional magnitudes................................. 79 3. Inversely proportional magnitudes............................... 82 4. Problems of compound proportionality..................... 84 5. Problems of proportional distribution......................... 86 Exercises and problems......................................................... 88 Maths workshop........................................................................ 92 Self-assessment........................................................................ 92

EQUATIONS

Page 132

1. Equations: meaning and use........................................... 134 2. Equations: elements and terminology......................... 136 3. Transposing terms............................................................... 137 4. Solving simple equations.................................................. 138 5. Equations with denominators......................................... 140 6. The general method for solving first-degree equations................................................................................ 141 7. Solving problems with equations.................................. 142 8. Second-degree equations................................................ 147 9. Solving second-degree equations................................. 148 Exercises and problems......................................................... 150 Maths workshop........................................................................ 156 Self-assessment........................................................................ 157

SYSTEMS OF EQUATIONS

Page 158

1. First-degree equations with two unknowns.............. 160 2. Systems of linear equations............................................. 162 3. Methods for solving linear systems.............................. 163 4. Solving problems with systems of equations........... 166 Exercises and problems......................................................... 171 Maths workshop......................................................................... 176 Self-assessment........................................................................ 177

PYTHAGOREAN THEOREM

Page 178

1. Pythagorean theorem........................................................ 180 2. Calculating a side when two are known..................... 182 3. Applications of the Pythagorean theorem................ 184 Exercises and problems......................................................... 187 Maths workshop........................................................................ 192 Self-assessment........................................................................ 193

SIMILARITY

Page 194

1. Similar shapes....................................................................... 196 2. Plans, maps and models.................................................... 200 3. How to build similar figures............................................. 202 4. Thales’ theorem.................................................................... 204 5. Similarity of right-angled triangles............................... 206 6. Applications of the similarity of triangles.................. 208 Exercises and problems......................................................... 210 Maths workshop........................................................................ 214 Self-assessment........................................................................ 215

GEOMETRIC SHAPES

Page 216

1. Prisms....................................................................................... 218 2. Pyramids.................................................................................. 220 3. Truncated pyramids............................................................ 222 4. Regular polyhedral.............................................................. 224 5. Plane sections of polyhedra............................................ 226 6. Cylinders.................................................................................. 228 7. Cones........................................................................................ 229 8. Truncated cones................................................................... 230 9. Spheres.................................................................................... 233 10. Sections of spheres, cylinders and cones................. 234 Exercises and problems.......................................................... 236 Maths workshop......................................................................... 242 Self-assessment......................................................................... 243

MEASURING VOLUME

Page 244

1. Units of volume..................................................................... 246 2. Cavalieri’s principle............................................................. 248 3. Volume of a prism and a cylinder.................................. 249 4. Volume of a pyramid and a truncated pyramid....... 250 5. Volume of a cone and a truncated cone..................... 252 6. Volume of a sphere............................................................. 253 Exercises and problems......................................................... 255 Maths workshop........................................................................ 260 Self-assessment........................................................................ 261

FUNCTIONS

Page 262

1. The concept of function.................................................... 264 2. Increases, decreases, maximums and minimums.... 265 3. Functions shown in tables of values............................. 266 4. Functions from their equation........................................ 267 5. Proportional functions: y = mx....................................... 268 6. The slope of a line................................................................ 270 7. Linear functions: y = mx + n............................................. 272 8. Constant functions: y = k.................................................. 274 Exercises and problems......................................................... 275 Maths workshop........................................................................ 280 Self-assessment........................................................................ 281

STATISTICS

Page 282

1. Making a table and its graph........................................... 284 2. Location parameters.......................................................... 286 3. Dispersion parameters....................................................... 288 4. Position parameters............................................................ 291 5. Two-way tables..................................................................... 293 Exercises and problems......................................................... 294 Maths workshop........................................................................ 300 Self-assessment........................................................................ 301

CHANCE AND PROBABILITY

Page 302

1. Random events..................................................................... 304 2. Probability of an event....................................................... 306 3. Assigning probabilities to regular experiments....... 308 4. Some strategies for calculating probabilities........... 310 Exercises and problems......................................................... 312 Maths workshop........................................................................ 316 Self-assessment........................................................................ 317

Annex • Glossary.................................................................................. 318

PYTHAGOREAN THEOREM Reading and listening

Egyptians and Babylonians The Pythagorean theorem is an important geometrical concept. As you know, it describes the relationship between the squares of the sides of any right-angled triangle. More than 3 000 years ago, both the Egyptians and the Babylonians knew that certain triangles were right-angled and they used them to construct right angles. The Egyptians, for example, used triangles whose sides measured 3, 4 and 5 (which they considered sacred) to divide fields and build pyramids. Pythagoras In his youth, Pythagoras (6th century BCE) travelled to Egypt and Babylon, where he undoubtedly learnt about these properties. His great achievement was to come up with the general theorem describing the relationship between squares drawn on the sides of any right-angled triangle. That is why this theorem bears his name. Euclid's Elements However, it was not Pythagoras who proved the theorem, but Euclid, two centuries later. Euclid of Alexandria wrote Elements around the year 300 BCE. It is a set of 13 books in which he collected, organised and expanded upon all the mathematical knowledge of his time, providing it with a solid logical structure. In Book I he demonstrated what we now know as the Pythagorean theorem. 1 Use context clues to define the phrasal verb ‘come up with’.

Then, write a new sentence in your notebook using this phrasal verb in a different situation. 2 Using past simple and past continuous, write sentences about the

history of the Pythagorean theorem starting with Pythagoras and ending with Euclid.

188

Pythagorean theorem Pythagorean theorem

When the blue triangle is right-angled: A=B+C

The area of the large square is equal to the sum of the areas of the two small ones.

3 What is the area of A if B = 9 cm2 and C = 16 cm2?

Euclid’s proof To prove the Pythagorean theorem, Euclid drew a line perpendicular to the longer side of the triangle (the green line), dividing the large square into two rectangles. He showed that: B = A1 and C = A2. A1 B

A2 C

4 Look at the figure and answer.

Taking a square from the grid as a unit: a) H ow many unit squares does the small square, B, contain? And rectangle A1? What do you notice?

A1 A2

9 B

b) C heck that the number of unit squares in C is the same as in A2.

c) Describe this property.

d) W hat theorem does it confirm? State the theorem.

ANK ANK B E G A U LANG LANGUAGE B ANK ANK GE BANK B B E E G G A A U U GUA K LANG ANG N L A L AN GE BANK 189 ANK GE BANK B B E E G G A A U U G A LAN LANG LANGUA LANGU

5 Draw a right-angled triangle with sides of 5 cm, 12 cm and 13 cm. Write

an equality expressing the relationship between the lengths of the sides.

PYTHAGOREAN 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. In general, we say that a is the hypotenuse and b and c are the legs. According to the Pythagorean theorem: a 2 = b 2 + c 2

This means that the area of the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs.

This relationship is only true if the triangle is right-angled. a 2 = b 2 + c 2 Interesting fact! This observation was made by the Chinese 400 years before Pythagoras was born. anayaeducacion.es GeoGebra. Graphical demonstration of the Pythagorean theorem.

Look at this demonstration:

The two identical squares have b + c as sides.

If we compare both figures, it is clear that a 2 = b 2 + c 2.

a2 a

a c

Problem solved

What are the areas of the unknown squares in the following shapes?

183 m2

102 m2

47 dm2

S1 386 dm2

As both triangles are right-angled, in both cases the area of the biggest square is equal to the sum of the areas of the smaller squares. Therefore: S1 = 183 m2 + 102 m2 = 285 m2 S2 = 386 dm2 – 47 dm2 = 339 dm2

ãã Pythagorean triples

F ocu s on Eng lish satisfy: to fulfill a condition.

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

Unit 9

ãã The sides of a triangle determine its type Look at the diagrams and analyse the explanation below: Remember

If we know the sides of a triangle, we can find out whether 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.

right-angled triangle

obtuse triangle

acute triangle

• We already know the relationship between the areas of the squares built on the sides of a right-angled triangle: Figure A 8 52 is equal to 32 + 42 8 In general: a 2 = b 2 + c 2 • If we expand the right angle making it obtuse, the length of the opposite side increases, as does the area of the corresponding square. Therefore: Figure B 8 62 is greater than 32 + 42 8 In general: a 2 > b 2 + c 2 • If we reduce the size of the right angle making it acute, the length of the opposite side decreases, as does the area of the corresponding square. Therefore: Figure C 8 4.52 is less than 32 + 42 8 In general: a 2 < b 2 + c 2 eas

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1 Copy these shapes in your notebook. Draw the square that is missing in each one and say what the area is.

a) b) A1

57 cm2

c) 3 dm2

87 m2

14 dm2

57 cm2

31 m2

A1 = 57 + 57 = … cm2 A2 = 87 + … = … m2 A3 = 14 – … = … dm2 2 Copy and complete to find out whether each of the following triangles is right-angled, acute or obtuse:

a) 70 cm, 240 cm, 245 cm

70 2 + 240 2 = 4 900 + 57 600 = 62 500 4 8 2452 < 702 + 2402 8 The triangle is… 245 2 = 60 025

b) 15 dm, 36 dm, 39 dm

15 2 + 36 2 = 225 + 1296 = f 4 8 392 … 152 + 362 8 The triangle is… 39 2 = 1521

c) 18 m, 80 m, 83 m

18 2 + 80 2 = 324 + 6 400 = 6 724 4 8 832 … 182 + 802 8 The triangle is… 83 2 = f

191

CALCULATING ONE SIDE WHEN TWO ARE KNOWN If we know that a triangle is right-angled, and we know the length of two sides, we can use the Pythagorean theorem to calculate the length of the third side.

ãã Calculating the length of the hypotenuse from two known legs Example a?

a 2 = b 2 + c 2 8 a = b 2 + c 2

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.

ãã Calculating one leg when the other leg and the hypotenuse are known Example

b? c

a 2 = b 2 + c 2 8 b 2 = a 2 – c 2 8 b = a 2 – c 2

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. eas

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1 Copy and complete to find the unknown side in each of these triangles:

The unknown side is the hypotenuse, a. a 2 = 112 + 142 8 a = f 2 + f 2 = f + f = f = … The hypotenuse measures approximately … dm.

11 dm

14 dm

56 km

48 km

The unknown side is the leg, c. 562 = 482 + c 2 8 c 2 = 562 – 482 8 c = f – f = … The other leg measures approximately … km.

Let's practise! 1 Find the length of the unknown side of the following right-angled triangles where

a is the hypotenuse. Round your answer to two decimal places if you need to. a) c = 70 mm; a = 74 mm e) b) b = 15 cm; a = 25 cm 15 cm c) b = 14 m; c = 48 m d) b = 13 inches; c = 84 inches 36 cm

f) 12 cm 37 cm

anayaeducacion.es Calculating one side when two are known. 192

Unit 9

ãã Practical exercises with guided solutions Below are various problems which show how the Pythagorean theorem can be used in everyday situations. eas

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Copy and complete these problems. Check that you arrive at the solution given. 2 A zip line is being fitted between two trees that are 12 m apart. The wire will be

attached to one tree at a height of 10 m and to the other at a height of 1 m. How long should the wire be if it needs to be fully taut but with 10 % extra length so that it can be tied to the trees? We know the legs and we have to find the hypotenuse. l 2 = 92 + 122 = … + … = … 8 l = f = … m The length of the taut cable is … m. Adding 10 % 8 … · 1.10 = 16.5 m

10 m

l 1m

12 m

3 We want to use a ramp to get a wheelbarrow up a 1 m step. We have a plank of

d 2 = 2.62 – … = 6.76 – … = … 8 d = f = 2.4 m The foot of the plank should be 2.4 m from the step, or slightly less so that it can rest on top of it.

2.6 d

wood that is 2.6 m long. How far from the step should the ramp start? We know the hypotenuse and the vertical leg. We calculate the other leg.

4 A balloon is tied to the ground with a 20 m rope. It is being blown by the wind,

with the rope pulled taut, so that the balloon is directly above a point on the ground 8 m from where it is tethered. How high is the balloon above the ground? We calculate the length of the vertical leg. h2 = …2 – 82 = … 8 h = f = … m The balloon is … m high.

5 A ladder whose foot is 4 m from the wall reaches a height of 7.5 m. How far from l

8.5 m

7.5 m

the wall would the foot of the ladder have to be for it to reach a height of 8 m? We first need to calculate the length of the ladder. l 2 = 42 + 7.52 = 72.25 8 l = 72.25 = 8.5 m Now we can calculate the distance we are asked to find, d. d 2 = …2 – 82 = … – 64 = … 8 d = f = … m The foot of the ladder should be … m from the wall.

h 2 = 202 – … 8 h = f = … m The height of the attic is … m.

15 5m

h 4m

of his attic. Calculate d , then h. First, we calculate the distance d: d 2 = 52 – 42 = … 8 d = f = … m Now we calculate the height, h, which is the leg of a right-angled triangle. The hypotenuse measures 20 m and the other leg … m:

6 Álvaro has taken the following measurements to find the height, h,

18 m 12 m

anayaeducacion.es GeoGebra. Calculating the unknown side using the Pythagorean theorem. 193

APPLICATIONS  OF THE PYTHAGOREAN THEOREM Many polygons have some elements which are sides of a right-angled triangle. This allows us to associate them using the Pythagorean theorem, and to calculate the length of one of the sides when the other two are known.

eas

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1 The diagonal of a rectangle is 89 cm long, and one of the sides is 80 cm. Calculate

the area. The area of the rectangle with sides a and b is: A = a · b We start by calculating the other side: b = 89 2 – 80 2 = … = … The short side is … cm long. The area is: A = 80 · … = 3 120 cm2

2 The diagonals of a rhombus are 10 cm and 24 cm long. Find the perimeter.

89 cm

80 cm

5 cm

We start by calculating the length of one side:

12 cm

s = 12 2 + … 2 = … = …

Each side is … cm long. The perimeter is: P = 4 · … = 52 cm

3 The side of a rhombus is 6.5 m long and one of the diagonals is 5 m. Find the area.

The area of a rhombus with diagonals d and d' is: A = d · d' 2 We know one of the diagonals. Calculate the other one using the Pythagorean theorem: d' = 6.5 2 – 2.5 2 = … m 2 The second diagonal is … · 2 = … m. Therefore, A = 5 ·… = 30 m2. 2 4 The bases of a right-angled trapezium are 25 cm and  38 cm long, and the height is

19 cm. Find the perimeter. We start by calculating the length of the oblique side: 13 2 + 19 2 =

… ≈… x= The oblique side is approximately … cm long. The perimeter is: P = 38 + 19 + 25 + … = 105 cm

2.5 m

6.5 m

d'/2

25 cm

19 cm 38 cm

x 13 cm

5 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 We start by calculating the height, h. The short side of the green triangle is (48 – 30): 2 = 9 cm. h = 41 2 – 9 2 = … = … The height of the trapezium is … cm. A = (30 + 48) · … = 1 560 cm2 2

b = 30 cm h

41 cm 9 cm

b' = 48 cm

anayaeducacion.es GeoGebra. Calculating the unknown side using the Pythagorean theorem. 194

Unit 9

6 Calculate the area of an equilateral triangle with 8 cm sides.

We begin by calculating the height: h = 82 – 42 = … ≈ … The height is approximately … cm. The area is: A = … · … = 27.6 cm2 2

8 cm

4 cm

7 Calculate the area and perimeter of a regular pentagon with an apothem of 16.2 cm

and a radius of 20 cm. We first calculate the side:

s = … 2 – … 2 = 137.56 ≈ … 2 The side of the pentagon is: s = … · 2 = … cm Therefore, its perimeter is: P = … · 5 = 117 cm Lastly, we calculate the area. Perimeter · apothem … · … = A= = 947.7 cm2 2 2

s/2 16.2 cm

20 cm

8 Find the perimeter 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 shortest side of the coloured right-angled triangle is: k = 6.6 : 2 = 3.3 cm Therefore: r = … 2 + … 2 = 42.25 = … The radius is … cm. P = 2πr = 2 · 3.14 · … ≈ 40.8 cm A = πr 2 = … · …2 ≈ 132.7 cm2

5.6 cm r

6.6 cm

9 A circumference with a radius of  8 cm is intersected by a line at two points, A and

B, which are 8 cm apart. Calculate the area of the circular segment bounded by the % arc AB . The pink circular segment is the difference between the circular sector that intercepts % the arc AB and the triangle OAB. The equilateral triangle OAB is the same triangle whose area we calculated in exercise 6 on this page (Atriangle = 27.6 cm2). 8c m O

m 8c

% Since it is an equilateral triangle, AOB = 60°. Therefore, the area of the sector is one sixth of the area of the whole circle. Asector = (π · …2) : 6 = … cm2 Therefore: Acircular segment = … – … = 5.9 cm2

8 cm

195

3 APPLICATIONS OF THE PYTHAGOREAN THEOREM eas

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10 The distance of a point P to the centre O of a circumference is OP = 23 cm. We

18 cm

r O

23 cm

r 2 = OP 2 – PT 2 = …2 – …2 = … Acircle = πr 2 = … · … ≈ 643.7 cm2 10 cm

11 Calculate the radius of the circumscribed circumference of this square with sides of

10 cm. Remember that the diagonals of the square are perpendicular. Therefore, the coloured triangle is right-angled and has legs of equal length. r 2 + r 2 = …2 8 2r 2 = … 8 r 2 = …

r = f = 7.07 cm D

12 Find the diagonal of a cuboid with dimensions of 1.2 m, 1.6 m and 4.8 m.

Looking at the blue triangle: AC = 1.2 2 + … 2 = … m

1.2 m

Looking at the red triangle: d = … 2 + 4.8 2 = … m The diagonal of the cuboid measures 5.2 m. Note that you can calculate it directly:

B D 4.8 m

d = 1.2 2 + 1.6 2 + 4.8 2 = f = 5.2 m

4.8 m

In general, in a cuboid with dimensions a Ò b Ò c the diagonal is: d = a 2 + b 2 + c 2

1.6 m

1.2 m 1.6 m

13 Calculate the height, h, of a regular pyramid that has a square base with  30 cm

h h

15 m

30 m 30 m

The pyramid is 8 m high. Let’s practise! 1 Find the area of an equilateral triangle with a perimeter

4 A 13-metre chord is drawn on a circumference with

2 Find the area and perimeter of an isosceles trapezium

5 A regular pentagon is inscribed in a circumference

of 54 cm.

with 3.2 m and 6.4 m bases, and 6.3 m height.

3 Calculate the area of a regular hexagon with 18 cm

sides. (Remember that in a regular hexagon the side and the radius have the same length.)

a radius of 9.7 m. How far is the line from its centre?

with a radius of 1 m. Its perimeter is 5.85 m. Calculate the area.

6 Find the length of the diagonal of a cuboid with sides

of 8 dm,  6 dm and 14 dm.

anayaeducacion.es GeoGebra. Calculating areas applying the Pythagorean theorem. 196

r to choose Remembe rtfolio. po ur yo r fo

resources

from this un

Unit 9

MS ES AND PROBLE

EXERCIS

Pythagorean theorem 1

Calculate the area of the green square in each of the following cases: a)

60 m2

14 cm2 30 cm2