heinemann maths zone 9 - chapter 3

8/9/2019 Heinemann Maths Zone 9 - Chapter 3

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113

he name ‘Pythagoras’ is attached to themost famous mathematical theorem of all

time. Pythagoras’ Theorem has had more

published proofs than any other theorem in

mathematics, and Pythagoras’ ideas have

influenced human thought in music,

philosophy, religion and geometry. Yet we

don’t have a single document that contains

anything that he wrote. His followers were

sworn to secrecy, so we can’t be certain about

any of his discoveries. We don’t even know for

sure that Pythagoras deserves to have hisname associated with the world’s most

famous theorem. Researchers have shown that

Pythagoras’ Theorem was understood by the

ancient Babylonians centuries before

Pythagoras was born. Was he, at least, the

first to prove it? Maybe. We can’t be certain.

Can you find the man of mystery in Raphael’s

famous painting, School of Athens?

Starter 3e

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HEINEMANN MATHS ZONE 914

Prepare for this chapter by attempting the following questions. If you havedifficulty with a question, click on its Replay Worksheet icon on your Student

DVD or ask your teacher for the Replay Worksheet.

1 State whether the following triangles are equilateral, scalene or isosceles.

(a) (b) (c)

2 Round each of the following numbers to two decimal places.

(a) 45.789 (b) 12.2311 (c) 4.5495678356

3 Use your calculator to find the following, correct to two decimal places where appropriate.

(a) 122 (b) 552 (c) 37.52

4 Use your calculator to find the exact values of each of the following.

(a) (b) (c)

5 Use your calculator to find the following, correct to two decimal places where appropriate.

(a) (b) (c)

6 Solve the following equations.

(a) x + 5 = 11 (b) 144 + x = 225 (c) 45 – x = 12

7 Solve the following equations. Where necessary, express your answer as adecimal correct to two decimal places.

(a) x 2 = 36 (b) x 2 = 39 (c) c2 = 325

Worksheet R3.1e

Worksheet R3.2e

Worksheet R3.3e

Worksheet R3.4e

81 169 256

Worksheet R3.5e

65 658 321.45

Worksheet R3.6e

Worksheet R3.7e

hypotenuse

irrational number

Pythagoras

Pythagoras’ Theorem

Pythagorean triad

rational approximation

right-angled triangle

surds

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3 ● pythagoras’ THEOREM 115

A triangle which contains a 90° angle, known as a right angle, iscalled a right-angled triangle. The longest side of a right-angledtriangle, which is opposite the right angle, is called thehypotenuse.

When labelling triangles, we use capital letters for the cornersor vertices and lower case letters for the side lengths.

So triangle ABC , which can be written as ∆ ABC , has vertices A ,B and C with sides a , b and c opposite the corresponding vertices.It is usual to label the hypotenuse as side c.

The Theorem of Pythagoras

One of the leading mathematics scholars in ancient Greece was Pythagoras ofSamos (580–496 BC ). It is said that Pythagoras discovered the theorem thatbears his name by looking at the pattern of square floor tiles.

C B

A

a

cb

Measure the angles and determine whether each of the following triangles is a right-angled

triangle. If the triangle is right-angled, measure the lengths, in mm, of the hypotenuse and

the other two sides.

(a) (b)

Steps Solutions

(a) 1. Measure the three angles in the triangles with a

protractor.

(a) The angles are 40°, 110°

and 30°.

2. Decide whether the triangle is right-angled. It is not a right-angled triangle.

(b) 1. Measure the three angles in the triangles. (b) The angles are 90°, 50° and

40°.

2. Decide whether the triangle is right-angled. It is a right-angled triangle.

3. If the triangle is right-angled, measure the

lengths of the hypotenuse and the other two

sides.

The length of the hypotenuse is

25 mm. The lengths of the

other two sides are 15 mm and

20 mm.

23 mm

15 mm

worked example 1


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For example, consider the right-angled triangle ofsides 3 cm, 4 cm, and 5 cm for the hypotenuse.

Here, a = 3, b = 4 and c = 5.

Now c2 = 52 = 25 and

a2 + b2 = 32 + 42

= 9 + 16 = 25

Thus c2 = a2 + b2 holds for this triangle; that is, thesquare on the hypotenuse is equal to the sum of thesquares on the two shorter sides.

Pythagoras’ Theorem states that, forany right-angled triangle, the square

of the length of the hypotenuse will be

equal to the sum of the squares of the

lengths of the two shorter sides.

That is: c 2 = a 2 + b 2

a

bc

hypotenuse

c2 = a2 + b2

a c

c2

b2

a2

b

25 cm2

9 cm2

16 cm2

34

5

4

5

3

Use a ruler to carefully measure the lengths, in millimetres, of the unknown sides of the

following triangles and determine whether Pythagoras’ Theorem holds for each triangle.

Hint: You may need to allow for some measurement error.

(a) (b)

Steps Solutions

(a) 1. Measure each side. (a) Lengths: 30 mm, 40 mm,

50 mm

2. Determine which is the longest side, and call it

c. Let the other sides be a and b.

Let c = 50

a = 30, b = 40

3. Check if these values satisfy Pythagoras’

Theorem.c2 = 502 = 2500

a2 + b2 = 302 + 402

= 900 + 1600= 2500

4. State conclusions. Since c2 = a2 + b2,

Pythagoras’ Theorem applies.

(b) 1. Measure each side. (b) Lengths: 30 mm, 45 mm,

60 mm

2. Determine which is the longest side, and call it

c. Let the other sides be a and b.

Let c = 60

a = 30, b = 45

30 mm30 mm

worked example 2


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If you are given the lengths of the sides of a triangle you can test them withPythagoras’

Theorem.

Skills

1 Measure the angles and determine whether each of the following trianglesis a right-angled triangle.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

3. Check if these values satisfy Pythagoras’

Theorem.c2 = 602 = 3600

a2 + b2 = 302 + 452

= 900 + 2025= 2925

4. State conclusions. Since c2 ≠ a2 + b2,

Pythagoras’ Theorem doesn’tapply.

If Pythagoras’ Theorem works, then the triangle must be a

right-angled triangle.

exercise 3.1 Pythagoras’ Theorem and right-angled triangles

Worked Example 1e

Hinte

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2 (a) For each of the triangles you identified as right-angled in Question 1 ,measure the lengths, in millimetres, of the unknown sides. Then copythe table below and complete it for each triangle. Let c be thehypotenuse of the triangle; it doesn’t matter which of the other sides

you call a and b. Hint: You may need to allow for some measurementerror.

(b) Can you see a relationship between a2 , b2 and c2 for these right-angled triangles?

3 For each of the following triangles select the correct statement ofPythagoras’

Theorem from those provided.

(a) A a2 + b2 = c2

B a2 + c2 = b2

C b2 + c2 = a2D a2 + b2 + c2 = 0

E b + c = a

(b) A p2 = q2 + r 2

B q2 = p2 + r 2

C r 2 = p2 + q2

D p2 + q2 = r 2

E p = q2 + r 2

(c) A z 2 + y2 = x 2

B x 2 + z 2 = y2

C x 2 – y2 = z 2

D z 2 = x 2 + y2

E x 2 – z 2 = y2

(d) A p2 = m2 – n2

B n2 + p2 = m2

C p2 + m2 = n2

D n2 – m2 = p2

E n2 + m2 = p2

4 Use a ruler to carefully measure the lengths, in millimetres, of the sides ofthe following triangles and determine whether Pythagoras’ Theorem holdsfor each triangle. Hint: You may need to allow for some measurement error.

Triangle a b c a2 b2 c2

Worked Example 1e

Worksheet C3.1e

a

b

c

r

q

p

y z

x

mn

p

Worked Example 2e

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Applications

5 Test to see whether triangles with the following side lengths are

right-angled triangles.(a) 9 mm, 12 mm, 15 mm (b) 2 cm, 4 cm, 6 cm

(c) 2 m, 2 m, 4 m (d) 7 km, 24 km, 25 km

(e) 6 m, 8 m, 10 m (f) 18 mm, 24 mm, 30 mm

(g) 5 cm, 12 cm, 13 cm (h) 11 mm, 60 mm, 61 mm

(i) 16 m, 30 m, 34 m (j) 12 cm, 19 cm, 23 cm

(k) 8 m, 15 m, 17 m (l) 6 mm, 7 mm, 8 mm

Analysis

6 For each of the following, construct right-angled triangles to scale, and

then complete the table after finding the length of the hypotenuse, c , bymeasurement. What is the relationship between a2 , b2 and c2?

a b c a 2 b 2 c 2 relationship

(a) 3 cm 4 cm

(b) 12 cm 16 cm

(c) 60 mm 80 mm

(d) 7 cm 24 cm

(e) 24 mm 90 mm

Hinte

Worksheet C3.2e

Hinte

Hinte

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7 Carefully draw four more right-angled triangles, measure the three sidelengths and verify that these triangles satisfy Pythagoras’

Theorem(allowing for some errors of measurement).

7

Proofs of Pythagoras’ Theorem

Pythagoras’ Theorem has fascinatedmathematicians for hundreds of years.They have developed many proofs,

which use combinations of geometryand algebra. You will look at two of

these geometric proofs. You will needgraph paper, scissors, a ruler andpencils.

Proof number 1

Follow these steps to re-create one proof of Pythagoras’ Theorem.

(a) Draw a right-angled triangle with sides of lengtha , b and c. It may be convenient to let a = 8 cm,b = 6 cm and c = 10 cm.

(b) Construct a large square using two small squaresof side length a and b and four right-angled

triangles of side lengths a , b and c (and area ab ). What is the area of the large square, which hassides of length (a + b ) units? Note: (a + b ) × (a + b ) = a × (a + b ) + b × (a + b )

It's a great theorem Pythagoras,but everyone will laugh if youcall it the hypotenuse.

a

bc

a a

a

b

b

b

c

c

b2

a2

a

12---


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(c) Rearrange the triangles as shown. Remember thehypotenuse is length c. What is the area of the squaresurrounded by the four triangles?

The large square still has sides (a + b ) units long,so it has the same area as the previous large square.

Write an expression for the area of the largesquare as the sum of the areas of the four trianglesplus the area of the square. Simplify if possible.

(d) The areas of the squares from parts (b) and (c) are equal. Write this as an equation andsee if you can simplify this expression to obtain Pythagoras’ Theorem, c2 = a2 + b2.

Proof number 2: Perigal’s proof

Follow these steps to re-create Perigal’s proof of Pythagoras’

Theorem.(a) Draw any right-angled triangle and

construct a square on each side.Label the squares A , B and C asshown.

(b) Divide the middle-sized square B with a line through its centre. Make PQ parallel to the hypotenuse (thelongest side of the right-angledtriangle). Draw RS perpendicular to PQ.

(c) Cut the middle-sized square B into the fourpieces marked. Also cut out the smallestsquare, A , and the largest square, C.

(d) Rearrange square A and the four parts ofsquare B to make a larger square. Measure thesides of this larger square and compare withsquare C. What do you notice?

(e) Explain how this demonstrates Pythagoras’Theorem.

a

b

a

b

c

c

c2

square C

square A

P

S

Q

R

square B


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When using Pythagoras’

Theorem we are dealing with squares and squareroots. Is it possible for all three numbers involved in the calculation(i.e. a , b and c ) to be whole numbers?

Pythagorean triads

A group of three whole numbers that satisfy Pythagoras’Theorem is called a Pythagorean triad.

The most common triads are (3, 4, 5), (5, 12, 13),

(7, 24, 25) and (8, 15, 17) and multiples of these. If wemultiply each of the numbers in a triad by the samenumber we get another triad.

For example, multiplying the numbers in (3, 4, 5)by two gives the triad (6, 8, 10), while multiplying the(5, 12, 13) triad by five gives the triad (25, 60, 65).

3

45

A Pythagorean trio.

Using your knowledge of common Pythagorean triads, state the value of the unknown side

in each of the following triangles.(a) (b)

Steps Solutions

(a) 1. Check if the values of the shorter sides match

a common triad, ( a, b, c ).

(a) a = 8, b = 15, c = ?(8, 15, 17)

2. State the answer. Hypotenuse = 17 cm

(b) 1. Check if the values of the hypotenuse and one

of the shorter sides match a common triad,

( a, b, c ).

(b) a = 6 = 3 × 2 and

c = 10 = 5 × 2, so (6, ?, 10) is a

multiple ( × 2) of the common

triad (3, 4, 5).

2. Find the missing number in the triad by

multiplying the ‘missing’ common triad number

by the common multiple.

Thus, b = 4 × 2 = 8

3. State the answer. The ‘missing’ short side is 8 m.

15 cm

8 cm10 m

6 m

worked example 3


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Surds and approximations Unfortunately, Pythagoras’

Theorem doesn’t always give ‘nice answers’.For example, if a = 5 and b = 7, then

a2 + b2 = 52+ 72

= 25 + 49

= 74.

Thus c2 = 74, and so the hypotenuse has length . This number, whenexpressed as a decimal, will have an infinite number of decimal places with norecurring pattern. It is called an irrational number.

These types of numbers are called surds. Other surds include numbers

such as , , etc. We can write the answer in exact surd form or,especially for real-life situations, in decimal form using a calculator. This iscalled a rational approximation. In questions that follow in this chapter, we

will write these answers correct to two decimal places, unless given otherinstructions.

Don’t forget to include the appropriate units with your answer.

74

2 7 35

The only way to write the exact value of a surd such as is to write the surd itself.

That is, the exact value of is . The number given by a calculator is actually

a rational approximation to the exact value. It is usual to round this approximation to agiven number of decimal places (in our case, to two decimal places).

74

74 74

Find the value of c, correct to two decimal places if necessary, using Pythagoras’

Theorem,

given that:

(a) a = 12 and b = 16 (b) a = 10 and b = 17

Steps Solutions (a) 1. Substitute the given values into Pythagoras’

Theorem.(a) c2 = a2 + b2

c2 = 122 + 162

2. Evaluate and simplify the right-hand side. = 144 + 256= 400

3. Find the value of c by taking the square root of

the number.

c = 400= 20

(b) 1. Substitute the given values into Pythagoras’

Theorem.(b) c2 = a2 + b2

c2 = 102 + 172

2. Evaluate and simplify the right-hand side. = 100 + 289

= 389

worked example 4


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Skills

1 Using your knowledge of Pythagorean triads, find the value of thehypotenuse in each of the following.

(a) (b) (c)

(d) (e) (f)

2 Using your knowledge of Pythagorean triads, find the value of theunknown side in each of the following.

(a) (b) (c)

(d) (e) (f)

3. Find the value of c by taking the square root of

the number. You will need a calculator to find

this number.

c =

= 19.723 082 92

4. Give the answer correct to two decimal places. = 19.72

389

Units are not always specified in problems. In this case, your answer will simply be a

number without units attached.

exercise 3.2 Squares, square roots, surdsand approximations

Worked Example 3e

Hinte

9 m

12 m

30 mm

40 mm

5 cm

12 cm

20 cm

15 cm2 m

1.5 m

4 mm

7.5 mm

Worked Example 3e

Hinte

5 mm13 mm

45 m

51 m7 cm

25 cm

30 cm

34 cm

12.5 m

3.5 m

6.5 cm2.5 cm

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3 Find the value of c , correct to two decimal places if necessary, usingPythagoras’

Theorem.

(a) a = 10, b = 24 (b) a = 5, b = 8 (c) a = 9, b = 12 (d) a = 8, b = 16

(e) a = 12, b = 15 (f) a = 50, b = 120 (g) a = 7, b = 12 (h) a = 11, b = 13

4 (a) Using Pythagoras’ Theorem, if a = 13 and b = 20, then c is closest to:A 33 B 569 C 24 D 23 E 18

(b) The value of x in the right-angled triangle is:A 10.39B 13.42C 18.00D 180.00E 108.00

5 (a) Using Pythagoras’ Theorem, if a = 15 and c = 17, then b is closest to:A 22 B 64 C 8 D 2 E 23

(b) The value of a , given that b = 36 and c = 39 and a , b , c form aPythagorean triad, is:A 15 B 225 C 514 D 23 E 13

Applications

6 Determine which of the following are Pythagorean triads.

(a) (6, 8, 10) (b) (24, 45, 51) (c) (14, 48, 50) (d) (25, 60, 80)

(e) (10, 16, 28) (f) (20, 48, 52) (g) (10, 18, 22) (h) (12, 50, 53)

7 Toula has just baked a cake in the shape of a rectangle. She wants to divide the cake into two right-angled triangles bycutting along a diagonal of the cake. Toula is then going to placea cake ribbon around the outside of each of the two triangularpieces of cake. If the length of the cake is 32 cm and its width is24 cm, how much cake ribbon will she need to buy, giving youranswer to an appropriate accuracy.

Analysis

8 (a) Daisy claims ‘In each Pythagorean triad with no common factor, itappears that there is one even and two odd numbers, one number

with a factor of 5 and at least one prime number.’ Consider thefollowing triads to decide whether Daisy’s statement is true. If

you had to modify her claim, what would you change it into?

3, 4, 5 8, 15, 17 20, 21, 295, 12, 13 12, 35, 37 28, 45, 53

7, 24, 25 16, 63, 65 36, 77, 85

9, 40, 41 20, 99, 101 44, 117, 125

11, 60, 61 24, 143, 145 52, 165, 173

13, 84, 85 28, 195, 197

15, 112, 113 32, 255, 257

17, 144, 145

(b) Can you find the pattern in each column and then write down the

next triad in each column?

Worked Example 4e

Hinte

12

6 x

Hinte

Homework 3.1e

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The secret society

Is the theorem all there is to Pythagoras? The answer is a definite no.Pythagoras of Samos was born about 570 BC on the Greek island of Samos. He died

about 475 BC , although the whereabouts and circumstances of his death are not absolutelyclear. Pythagoras was well educated and may even have been taught by Thales, one of thegreatest teachers of ancient Greece. He travelled to Egypt where he was accepted into thepriesthood in the temple at Diospolis. During this time Pythagoras became familiar withmany customs and practices that he later imposed on his own society, the Pythagoreans.

Upon his return to the Mediterranean Pythagoras founded a philosophical and religiousschool at Croton, which was located on the east side of the heel of what is now Italy. ThePythagoreans had to obey some strict rules. As an example, new members were not allowedto speak for five years! Because of the secretive nature of the group and the communalnature of their work it is virtually impossible to tell which work was done by Pythagorashimself and what was done by other members of the society. However, we do know that

women were members. In fact, Pythagoras’ wife, Theano, was a member in her own right,and one of his daughters, Damo, was given the responsibility for keeping her father’s

writings secret after his death.Even though we call the formula above the Theorem of Pythagoras, the result it explains

was known for hundreds of years before Pythagoras. The pyramids have square bases formedusing a 3, 4, 5 triangle. A Babylonian tablet dated to 1900 BC contains a table of what we nowcall Pythagorean triads, and an ancient Chinese text, the Chou-pei, shows that the Chinese


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also knew about at least the triads many hundreds of years before the Pythagoreans. ThePythagoreans were probably the first to prove the theorem in a formal way.

The discovery of irrational numbers was an interesting one.The Pythagoreans had a motto: ‘All is number’. By this theymeant everything in life could be reduced to an analysis ofnumber, and numbers were always able to be expressed as the

ratio of two other numbers. In the pursuit of ‘triples’ thePythagoreans came across the right-angled isosceles trianglethat had its equal sides one unit long. The length of thehypotenuse of this triangle is the irrational number . There arestories that the Pythagoreans tried to keep this a secret as it went against their generalphilosophy, but no one knows if these are true.

The Pythagoreans also worked on number patterns. They gave names to certain familiesof numbers:• Perfect numbers have the sum of their proper divisors equal to the number itself. As an

example, the proper divisors of 6 are 1, 2, 3 and 1 + 2 + 3 = 6. (The proper divisors are all

the divisors except the number itself.)• Abundant numbers have the sum of their proper divisors greater than the number itself.

As an example, the proper divisors of 12 are 1, 2, 3, 4, 6 and 1 + 2 + 3 + 4 + 6 = 16.• Deficient numbers have the sum of their proper divisors less than the number itself. As

an example, the proper divisors of 8 are 1, 2, 4 and 1 + 2 + 4 = 7.• Amicable numbers are pairs of numbers a and b such that a is the sum of the proper

divisors of b and b is the sum of the proper divisors of a. The smallest pair of amicablenumbers is 220 and 284.

Questions

1 The expressions , a , , where a is an odd integer, generate Pythagorean triads.(a) Find the triads for a = 3, 5, 7, 9, 11.

(b) What happens if a = 1?

2 Show that the numbers 220 and 284 are indeed amicable.

3 The symbol of the Pythagorean society was thepentagram, or star pentagon.

Draw a regular pentagon on a plain sheet ofpaper. Make the sides 10 cm long. Use a protractorto mark the internal angles. They will be 108°. Then

draw the internal diagonals and label the diagramas shown. Now measure AC , AD' and D'C to thenearest millimetre.

(a) Find the ratios AC : D'C and D'C : AD' andconfirm that these are equal, allowing forsome measurement inaccuracy.

(b) Do a similar calculation for one of the other diagonals.

(c) What type of shape is A'B'C'D'E'?

Research

Make a poster presentation of at least five different proofs of thePythagoras’ Theorem.

1

1

2

a2 1–2--------------

a2 1+2--------------

A

B

CD

E

A'

B'

C' D'

E'

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It is possible to use Pythagoras’ Theorem to find the length of the hypotenuseof a right-angled triangle, if we know the lengths of the two shorter sides.

Pythagoras’ Theorem may be applied to other shapes provided they containright-angled triangles. Or you may need to solve a problem that will requirePythagoras’ Theorem even when there is no right-angled triangle in thediagram. In these cases it is necessary to find the missing right-angled triangle

by carefully adding a line or lines to the diagram.

Find the length of the hypotenuse, correct to two decimal places if necessary, in the

following right-angled triangle.

Steps Solution

1. State Pythagoras’ Theorem, defining the side

lengths.c2 = a2 + b2

a = 6, b = 10 and c = x .

2. Substitute the values into Pythagoras’ Theorem. x 2 = 62 + 102

3. Simplify the right-hand side. = 36 + 100 = 136

4. Take the square root of both sides to find the

value of x . You will need a calculator to find this

number.

x =

and so x = 11.661 903 78

5. State the length of the hypotenuse, correct to two

decimal places.

The hypotenuse has length 11.66

units.

10

x6

136

worked example 5

Although (+25)2 and (–25)2 both equal 625, only +25 makes sense as the side length of

a triangle. With Pythagoras’ Theorem use only the positive square root.


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Use Pythagoras’ Theorem to find the value of the pronumerals, correct to two decimal

places if necessary, in each of the following diagrams.

(a) (b)

Steps Solutions

(a) 1. Identify a right-angled triangle and define the

side lengths.

(a) Let the hypotenuse be c, and

a = 2, b = 4.

2. Substitute the values into Pythagoras’

Theorem.c2 = a2 + b2

c2 = 22 + 42



value of the unknown. You will need a

calculator to find this number.

c =

and so c = 4.472135 955

5. State the length of the unknown side, correctto two decimal places.

The diagonal has length4.47 m.

(b) 1. Draw a line from point B down to the line CD to

form a right-angled triangle BED.

(b)

2. Redraw the right-angled triangle and define the

side lengths.

Let the hypotenuse

be c = z , and

a = 10, b = 3.

3. Substitute the values into Pythagoras’

Theorem.c2 = a2 + b2

z 2 = 102 + 32


2 m

4 m

c m z

B12

DC

A

10

15

2 m

4 m

c m

20

z

B12

DC

A

10

15 3

z

B

DE

10

3

worked example 6


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Skills

1 Find the length of the hypotenuse in these diagrams.

(a) (b) (c)

(d) (e) (f)

2 Find the length of the hypotenuse, correct to two decimal places if

necessary, in these right-angled triangles.(a) (b) (c)

(d) (e) (f)

3 The length of the hypotenuse in this right-angled triangle, correct to twodecimal places, is closest to:A 39.80 mmB 63.51 mmC 88.00 mmD 4034.00 mm

E 1584.00 mm


value of the unknown. You will need a

calculator to find this number.

z =

= 10.440 306 51

6. State the length of the unknown side, correct

to two decimal places.

The unknown side has length

10.44 units.

109

exercise 3.3 Finding the hypotenuse of a right-angled triangle

Worked Example 5e

Animatione

x

12

5 x

6

8

x

9

12

x

36

15

24

x10

x

60

25

Worked Example 5e

Hinte

Worksheet C3.3e

4

x13 x

12

6 x

4.8

3.8

x

14.7

8

8 cm

x3 cm

16 mm

14 mm x

53 mm

35 mm

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4 A ladder is leaning against the wall of a schoolbuilding and reaches just above the guttering, a heightof 4.27 m above the ground. The foot of the ladder is1.41 m from the base of the building. What is thelength of the ladder, correct to two decimal places?

5 The value of y in the following diagrams is:

(a) A 6.31

B 3.2

C 73.12D 7.3

E 36.48

(b) A 32.57

B 21

C 25.64

D 31

E none of the above

6 Find the value of the unknown side, c , in each of these diagrams.

(a) (b) (c)

(d) (e) (f)

Worked Example 5e

y

4.1

6.04

10

29.34

y

Worked Example 6e

Hinte

8 cm

15 cm

c cm 30 m

16 m

c m

c cm

12 cm

6 cm

8 cm

c m

4 m

4 m

7 m28 m

21 m c m

c cm

12 cm

8 cm

18 cm

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7 Find the value of c , correct to two decimal places, in these diagrams.

(a) (b) (c)

(d) (e) (f)

8 The value of the x in the diagram is closest to:A 16.4 cmB 17.7cmC 15.6 cmD 27.0 cmE 28.2 cm

Applications

9 Find the value of the pronumerals, correct to two decimal places ifnecessary, in the following diagrams.

(a) (b) (c)

(d) (e) (f)

Worked Example 6e

Hinte

7 cmc cm

7 cm

15 mmc mm

23 mm

c cm

17 cm

5 cm

15 cm

c m

14 m

10 m

10 m

c m

3.2 kmc km

4.5 km

c m

12 m

5 m

12 m

x cm

25 cm

10 cm

13 cm

x cm

Hinte x cm

12 cm 6 cm

8 cm y cm x cm

8 cm

10 cm y cm

7 cm

a cm

10 m

b cm

5 m

18 m

4 m

x mm

y mm

16 mm

20 mm

12 mm

3 cm

5 cm

a cm

b cm

6 cm

b cm

a cm

4 cm7 cm

c cm

4 cm

4 cm

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10 The values of x and y in the diagram are closest to:A x = 10.39 cm, y = 7.21 cmB x = 7.21 cm, y = 13.42 cmC x = 13.42 cm, y = 7.21 cmD x = 13.42 cm, y = 4.47 cmE x = 10.29 cm, y = 4.47 cm

Analysis

11 (a) For a right-angled isosceles triangle, with the two equal shorter sideshaving length 10 cm, find the length of the hypotenuse. Now dividethe length of the hypotenuse by the length of the shorter side. Do yourecognise this number?

(b) Find the length of the hypotenuse for several other right-angledisosceles triangles. Then divide the length of the hypotenuse by thelength of the shorter side. What do you notice about the result?

12 A swimming pool is 50 m long and

25 m wide. Anastasia is trying to swim alength of the pool, starting from acorner at the shallow end. However shehas trouble with her direction, andusually ends up somewhere betweenthe two corners at the other end of thepool.

(a) Draw a diagram and calculate fourpossible distances, correct to twodecimal places, that Anastasia

may actually end up swimming bycompleting her lap of the50 m pool.

(b) What is the maximum distance Anastasia can actually swim whencompleting a lap of the pool?

There is one other type of problem that needs Pythagoras’

Theorem for itssolution. The length of one of the shorter sides of a right-angled triangle canbe found if we are given the lengths of the hypotenuse and the other side.

x cm

12 cm 4 cm

y cm

Hinte

12


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Remember: if you are finding a side other than the hypotenuse, then it must beshorter than the hypotenuse.

It may be necessary to draw out a right-angled triangle from a diagram in orderto apply Pythagoras’

Theorem.

Find the value of y , correct to two decimal places if necessary, in this

right-angled triangle.

Steps Solution



a = y , b = 14, c = 19

2. Substitute the values into Pythagoras’ Theorem. 192 = y 2 + 142

3. Rewrite so that y 2 is the subject of the

expression. y 2 = 192 – 142

4. Simplify the right-hand side. = 361 – 196 = 165


value of y. You will need a calculator to find this

number.

y =

y = 12.845 232 57

6. State the length of the unknown side, correct to

two decimal places.

The side has length 12.85 units.

y

1419

165

worked example 7

Find the value of x , correct to two decimal places, in this

diagram.

Steps Solution

1. Draw a diagram of the appropriate right-angled

triangle.



Let a = 10, b = x and c = 12

12 m

10 m

x m

12

10

x

worked example 8


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Skills

1 Find the value of the unknown shorter side in these diagrams.

(a) (b) (c)

(d) (e) (f)

2 Find the value of the unknown shorter side, correct to two decimal places,in these diagrams.

(a) (b) (c)

(d) (e) (f)

3. Substitute the values into Pythagoras’ Theorem. 122 =102 + y 2

4. Rewrite so that x 2 is the subject of the

expression. x 2 = 122 – 102

5. Simplify the right-hand side. = 144 – 100= 44

6. Take the square root of both sides to find thevalue of y. You will need a calculator to find this

number.

y =

y = 6.633 249 581

7. State the answer correct to two decimal places. The unknown side is x = 6.63 m.

44

exercise 3.4 Finding a shorter side of a right-angled triangle

eTutoriale

Worked Example 7e

Animatione

1237

25

7

12

15

41 m

9 m

20

52

85 mm 36 mm

Worked Example 7e

Hinte

y

1429

y

8

21.3

y

326

Worksheet C3.4e

127

45

y

15 m

y

10 m

11.2 m20.4 m

y

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3 A playground slide hasa length of 4.76 m, whileit covers a distance of4.10 m on the ground.

What is the height ofthe slide, correct to two

decimal places?

4 (a) The value of x in this right-angledtriangle is closest to:A 12.99 B 17.13 C 23.10D 7.30 E 14.25

(b) The hypotenuse of a right-angledtriangle has length 55 cm. One of the shorter sides has a length of53 cm. The length of the third side is closest to:A 2 cm B 216 cm C 14.6 cm D 14.7 cm E 15.0 cm

Applications

5 Find the value of the pronumerals, correct to two decimal places ifnecessary, in each of the following diagrams.

(a) (b) (c)

(d) (e) (f)

6 Find the lengths of the shorter sides, correct to two decimal places, if thelength of the hypotenuse of a right-angled isosceles triangle is as given.

(a) 20 cm (b) 12 m (c) 100 cm (d) 35 m

Analysis

7 Find one set of possible lengths of the shorter sides, correct to two decimalplaces, if the length of the hypotenuse of a right-angled scalene triangle isas given.

(a) 20 cm (b) 12 m (c) 100 cm (d) 35 m

4.10 m

4.76 m

?

Hinte

Worked Example 7e

x

15.27.9

Worked Example 8e

Hinte

x m15 m

12 m

x m

10 m

14 m

x m

10 m

6 m

5 m

x mm

28 mm

34 mm

10 mm

x cm

y cm

10 cm

15 cm

1 3 c m

y m

20 m12 m

x m

10 m

Hinte

Worksheet C3.5e

7

eQuestionse

Homework 3.2e

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Answer the questions, showing your working, and then arrange the letters in the order shown

by the corresponding answers to find the cartoon caption.

Find each side length in cm.

Complete the Pythagorean triad (6, 8, E ). E

The length of the hypotenuse, G , of a right-angled triangle if the two shorter sides Gare 15 cm and 20 cm.

Complete the Pythagorean triad (7, H , 25). H

The length of the unknown side of a right-angled triangle, I cm, if the hypotenuse is I

17 cm and the known short side is 8 cm.

The length of the N The length of the O The length of the P

hypotenuse in this triangle: hypotenuse in this triangle: unknown shorter side:

The length of the S The length of the T The length of the U

unknown shorter side: unknown shorter sides: unknown shorter sides:

24 15 25 24 19.2 25.5 17.7 15 21.2 26.9 23.2 10

N cm7 cm

20 cm O cm18 cm

18 cm

P cm27 cm

19 cm

S cm30 cm

19 cm

T cm

T cm

25 cm U cm

U cm

38 cm

‘ ’


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Pythagoras’

Theorem is a powerful mathematical tool, having uses ingeometrical proofs as well as in many practical situations.

Solving problems involving

right-angled triangles

Right-angled triangles are found in many situations, and so Pythagoras’Theorem is often useful for calculations based around these triangles. Withpractical problems, follow these steps:

Problems may involve finding the hypotenuse.

1 Draw a diagram (involving a right-angled triangle) to illustrate the

situation described in the problem statement (if not already given).

2 Label key points of the diagram such as the right angle and the

hypotenuse, and the measurements specified in the problem.

3 Label the measurements you are requested to find with

pronumerals.

4 If necessary, re-draw the appropriate right-angled triangle.

5 Use Pythagoras’ Theorem to find the required measurement.6 Make sure that you answer the problem as required (to the required

accuracy, along with appropriate units). If it is a sentence question,

answer in a sentence.

What is the length of the diagonal support of thehouse frame shown? State your answer correct to

two decimal places.

3 m

6 m

worked example 9


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Alternatively, problems may involve finding the length of one of the shortersides.

Steps Solution

1. Draw a diagram of the appropriate

triangle.

2. Use Pythagoras’ Theorem to find the

value of the pronumeral.c2 = a2 + b2

c2 = 62 + 32

= 36 + 9 = 45

c =

Screen shows 6.708203 932

3. State the answer correct to two

decimal places.

The diagonal is 6.71 m long.

3 m

6 m

45

A ladder 3.5 m long is leaning against a vertical wall with the base 1.5 m from the bottom of

the wall on horizontal ground. How high up the wall does the ladder reach, correct to two

decimal places?

Steps Solution

1. Draw a labelled picture with all

measurements indicated.

2. Draw a diagram of the appropriate

triangle.

3.5 m

1.5 m

A

B C

3.5 m

1.5 m

A

B C

x

worked example 10


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Where necessary, state your answers correct to two decimal places. Skills

1 A support wire is attached 5 m up a flagpole. Theother end is attached to the ground 3 m from thebase of the flagpole. How long is this wire?

2 The escalator at the localdepartment store is 18 m long.

When you ride on the escalator you move across 15 m. Theheight through which youtravelled is:A 9.95 m B 23.43 mC 12.28 m D 3 mE none of the above

3 Andrew is standing at the corner of thepark, and has decided to cross diagonallyrather than go around.

(a) How far does Andrew travel whencrossing the park?

(b) How much distance does he save bynot going around?

3. Use Pythagoras’ Theorem to find the

value of the pronumeral.

c2 = a2 + b2

Let a = 1.5, b = x , c = 3.5

3.52 = 1.52 + x 2

x 2 = 3.52 − 1.52

= 12.25 − 2.25

= 10 x =

x = 3.16

4. Give your answer in the context of the

question.

The ladder will reach 3.16 m up the wall.

10

exercise 3.5 Applications of Pythagoras’Theorem

3 m

5 m x

Worked Example 9e

Hinte

18 m

15 m

x

Hinte

28 m

35 m

Worked Example 9eHinte

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4 A 13 m long ladder is resting against a wall with its foot 5 m from the baseof the wall. How high up the wall does the ladder reach?A 10 m B 12 m C 18 m D 14 m E 11 m

5 Kensington race course has races thatare run over a straight 1200 m course.One horse starts near the outside fence

and runs in a straight line towards thefinishing post on the inside fence. If thetrack is 35 m wide, how much furtherdoes this horse run than one that runs ina straight line along the inside fence?

6 The bottom of a slide is 3.6 m from the base ofthe vertical ladder. If the slide is 5 m long, howhigh is the ladder?

7 Edmond has attached a 17 m long support wire to a radio tower 2 m fromthe top of the tower. If the other end of the wire is attached to the ground7.5 m from the base of the tower, the total height of the tower is:A 17.26 m B 15.26 m C 18.58 m D 20.58 m E 19 m

8 A flagpole, 10 m tall, is supportedby two wires, each of length 16 m

which are fastened to bolts in the

ground. How far apart are thepegs?A 156.00 m B 12.49 mC 24.98 m D 16.97 mE 33.94 m

9 The length of the hypotenuse on an isosceles right-angled triangleis 16 cm. What is the length of the other two sides?

10 During a heavy rainstorm, the road that Aysenormally takes to travel home from school hasbecome flooded. She must take a detour as

shown in the diagram. How much furtherdoes she now have to travel?

1200 m

35 m x

Worked Example 9e

3.6 m

5 m x

Worked Example 10e

16 m10 m

16 m

Hinte

Worked Example 10e

Hinte

flood

3.6 km

2.9 km

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Applications

11 The end view of a garden shed is shownin the diagram. The shed is 5.2 m wide,the sides are 2.1 m high and the slopingpart of the roof is 2.8 m. Calculate theoverall height of the shed, to anappropriate level of accuracy.

12 Two steel discs, one with diameter 6 cmand one with diameter 8 cm, are placed in a square

wooden frame. Find the length of the sides of thisframe.

13 The roads corporation needs tobuild a new road between thetwo towns of Alderton andZincton. Unfortunately there isa mountainous region betweenthe towns. It costs $2500 perkilometre to build a roadaround the mountains and$4000 per kilometre to build a road through the mountains. Which is thecheapest way to build the required road?

14 The frame for the wall of a new building is made from steel girders that are

10 m long. The frame is rectangular, 7.5 m high and 18 m long, and it has onediagonal support. Find the cost of the frame if the beams cost $230.00 each.

Analysis

15 Jenny is standing at A , the corner ofseveral paddocks. She wishes to travel tothe points B , C , D , E and F , and finishback at A after travelling the shortestpossible distance. In what order shouldshe visit each point, and what will be thedistance travelled?

16 Angela has built a new rectangulargate from timber. The gate is 1.5 mhigh and 3.1 m wide. She has decidedto put in two diagonal cross supportsto give the gate more strength. What isthe total length of extra wood she willneed?

5.2 m

2.1 m

2.8 m

Animatione

18 km

35 km A

Z

20 m

15 m

30 m

10 m A

B

F

C

D

E

Hinte

3.1 m

1.5 m

http://anim/PyProb.exehttp://anim/PyProb.exehttp://anim/PyProb.exe


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The cost of the gate depends on the amount of timber it takes to make it. Would it be cheaper to make the gate from one of these designs?

Design your own gate to be the same height and width as Angela’s.

How does the relative cost of your gate compare?

Hinte

Homework 3.3e

Restarter 3e

Fire rescues

Investigating and designing

Fire trucks with extension ladders are used regularly toremove people from burning buildings. However, theyare limited in how far they can reach, by both the lengthof the ladder and, for stability reasons, the distance theycan safely park away from the edge of the building.

Assume that the ladder is at the back of the truck and itsfoot is 2 m off the ground.

1 Draw a diagram to show a truck with its ladder anda building. Label the unknown distances withpronumerals.

Producing

The truck is 4 m away from the base of the building andthe fully extended ladder reaches 16.46 m up the side ofthe building.

2 Use your diagram to find the length of the ladder tothe nearest metre.

3 It is decided that the trucks must park between 2 mand 7.5 m from buildings. What is the maximum height that the ladder can reach from eachextreme position?

4 How far from the wall is the truck if the fully extended ladder reaches a height of 15.5 m?

Analysing and evaluating

5 A building has its windows 1.5 m above the ground and then the windows of successivefloors are 4.2 m apart vertically. Which would be the highest floor that the ladder couldreach?

6 For Question 5 , what length of ladder would be needed to reach to the 6th floor?

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DIY summary

Copy and complete the following using the words and phrasesfrom the list where appropriate to write a summary for this chapter.

A word or phrase may be used more than once.

1 Why is a right-angled triangle so named?

2 The longest side of a right-angled triangle is called the________.

3 For any _________ triangle, the square of the length of the

__________ is equal to the sum of the squares of the lengths ofthe two shorter sides.

4 The equation c2 = a2 + b2 is known as __________ ______. It isnamed after ___________, the person credited with developingthis ‘formula’.

5 A group of three whole numbers that satisfy Pythagoras’Theorem, for example (3, 4, 5), is called a ____________ ____.

6 Numbers such as , , are called _______.

7 What sort of number is a surd?

8 We usually write the value of numbers such as as a___________ ________________.

VELS personal learning activity

1 Draw a right-angled triangle and label all sides. Write down Pythagoras’ Theorem using the labelsthat you have used.

2 Show how Pythagoras’ Theorem works using a Pythagorean triad.

3 Make up a little rhyme, rap or tune to assist you to remember Pythagoras’ Theorem. Share it witha partner.

4 Explain how you could work out if a triangle was right-angled if you don’t have a protractor butdo have a ruler.

5 Explain what ’rational approximation’ means and why it is needed. Give an example.

hypotenuse

irrational number

Pythagoras

Pythagoras’ Theorem

Pythagorean triad

rational approximation

right-angled trianglesurds

2 7 35

84

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Skills

1 Measure the angles and determine whether each of the following trianglesis a right-angled triangle.

(a) (b) (c)

2 Use a ruler to carefully measure the lengths, in millimetres, of the sidesof the following triangles and determine whether Pythagoras’

Theoremholds for each triangle. (You may need to allow for some measurement error.)

(a) (b)

(c)

3 By substituting into Pythagoras’ Theorem, test to see whether triangles with the following side lengths are right-angled triangles.

(a) a triangle with sides 6 cm, 11 cm, 14 cm

(b) a triangle with sides 14 mm, 48 mm, 50 mm

4 Using your knowledge of common Pythagorean triads, state the value ofthe unknown side in each of the following triangles.

(a) (b) (c)

5 Find the value of c , correct to two decimal places if necessary, usingPythagoras’ Theorem given that:

(a) a = 45 and b = 60 (b) a = 32 and b = 22 (c) a = 7.5 and b = 10

3.1

40 mm30 mm

40 mm

35 mm

3.1

3.1

3.2

12 cm

16 cm

48 m

14 m 34 m16 m

3.2


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Applications

11 Rebecca is mounting a new 3.0 m antenna on the roof of her house. Sheneeds to hire a ladder which will need to extend 1.2 m above the level ofthe guttering, which is 4.2 m above the ground as shown in the diagram.

(a) If the antenna is placed on the roof 0.9 mhorizontally from the outer wall, what isthe horizontal distance between the footof the ladder and the foot of the antenna?

(b) Hence calculate what length ladder (to thenearest cm) Rebecca should hire.

Once she has mounted the antenna on theroof Rebecca will need to attach a support

wire connected to the top of the antenna,7.5 m above ground, to a peg in the ground.

(c) If the wire is 9.0 m long, how far from the wall should the peg beplaced?

12 Ming and Tong have decided tohave a race from the gate to the

playground in Macalister Park.Ming has decided to run alongthe left edge of the path and cando so at a speed of 7 metres persecond. On the other hand, Tongdecides to go directly from thegate to the playground but this is very sandy and he can only run at5 metres per second. Who wins the race to the playground?

13 A playground slide is 2.6 m high.The ladder is 3.6 m long. The

distance from the bottom of theslide,C , to the base of the ladder, A , is 6.8 m. The end of the slide,CD , is 2 m. Find the total lengthof the slide from B to D.

14 Tony was to meet Lynne at thelocal park. What would be theshortest distance for Tony totravel from his position at point A to Lynne’s position at point B?

3.5

2.9m

3.0m

4.2m

1.2m

Gate

65 m

140 m

P l a y

g r o u n d

3.5

3.6 m

6.8 m

B

A DC

2.6 m

2 m

3.5

55 m

25 m

15 m

70 m

70 m

95 m

A

B 3.5


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Analysis

15 Daniel and Courtney are going to make a garden in an area in thebackyard of their new house. They have measured the space and itis shown in the diagram.

(a) Calculate the perimeter of the garden and find the number of2 m long sleepers they will need to form the perimeter of thegarden.

(b) If 2 m long sleepers cost $35 each, how much will it cost toplace sleepers all around the garden bed?

16 Before each horse race, a barrier draw occurs to determineeach horse’s starting position. Closest to the rails is the moredesired position and is represented by point A on the diagramshown. The least preferred position is near the outside edgeof the track (point B). It can be assumed that all jockeys aimto take their horses in a direct line to the beginning of the firstcorner (point C).

(a) How much further does the horse at point B travel than the horse atpoint A?

(b) Is this difference significant? Comment on this.17 Four steel discs, each with radius 1 cm, are placed in

a square frame.

(a) Find the length of the sides of this frame.

(b) In order to brace the frame, a diagonalsupport is to be added to the frame. Find thelength of this support.

(c) Find the length of material needed to buildthe frame, including the diagonal brace.

1 Calculate:

(a) −87 − 110 (b) 120 + 45 − 200 (c) −5 + 10 × 7

2 Solve for x :

(a) −7 x = 35 (b) 17 − x = (−30) (c) x + 45 = (−10)

3 A cube has a side length of 4 cm. Find its volume.

2 m

5 m

4 m2 m

C

B

A

400 m

40 m

startingstalls

1 cm 1 cm

1 cm

1 cm

Hinte

Worksheet R3.8e

Worksheet R3.9e

Worksheet R3.10e

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heinemann maths zone 9 - chapter 3

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