NSM_8_CB_Ch03

3 Investigating Pythagoras’ Theorem We Egyptians first discovered the {3,4,5} right-angled triangle long before Pythagoras.

... I took the idea further and put my name on it - long before copyright laws!

c2 = a2 + b2

Chapter Contents 3:01 Investigating right-angled triangles Investigation: Right-angled triangles Investigation: Pythagoras’ Theorem 3:02 Using Pythagoras’ Theorem: Calculating the hypotenuse Investigation: Pythagorean triads 3:03 Using Pythagoras’ Theorem: Calculating one of the short sides

MS4·1

MS4·1

Fun Spot: What happens when ducks fly upside down? 3:04 Miscellaneous exercises involving Pythagoras’ Theorem MS4·1 Investigation: Pythagoras and speed Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically

MS4·1

Learning Outcomes MS4·1

Uses formulae and Pythagoras’ Theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles. Working Mathematically Stage 4 1 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting

32

The Egyptians made use of the fact that a triangle that had sides 3 units, 4 units and 5 units in length was always a right-angled triangle. However, it was a religious group led by the Greek philosopher Pythagoras, around 550 BC, that discovered the relationship which exists between the sides of all right-angled triangles.

3:01 | Investigating Right-angled

Outcome MS4·1

Triangles

As we investigate right-angled triangles we will call the side lengths a, b and c, where c will always stand for the length of the longest side. We call the longest side of a right-angled triangle the hypotenuse.

c

a

b

c

3 cm

4 cm

5 cm

The longest side of a right angled triangle is called the ‘hypotenuse’.

b The hypotenuse is opposite the right angle. a

In triangle ABC (also called ∆ABC), BC = 4·5 cm, CA = 6 cm and AB is 7·5 cm. • This is also a {3, 4, 5} triangle if we let each unit be 1·5 cm. The side lengths are: 3 × 1·5 cm, 4 × 1·5 cm and 5 × 1·5 cm. • Is ∠ACB a right angle? Which side is the hypotenuse?

B

c a

C

b

a

b

c

4·5 cm

6 cm

7·5 cm

3 × 1·5 cm

4 × 1·5 cm

5 × 1·5 cm

A

investiga

tion

Investigation 3:01A | Right-angled triangles 1 Take three lengths of rope or string 3 m, 4 m and 5 m in length. Use them to form a triangle and measure the largest angle. Is it a right angle?

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

3:01A

33

2 Measure 1·5 m up and 2 m along the bottom of a wall that stands on flat ground. How long is the distance between these two points? Use string to measure this distance. Is the triangle a {3, 4, 5} triangle?

3 Draw or find some right-angled triangles in the school grounds. Make the sides any length you wish. For each triangle, measure the side lengths a, b and c (where c is the longest side) and enter these in a table like the one below. Make sure you include the three already in the table. a

3

6

5

b

4

8

12

c

Exercise 3:01 1

Foundation Worksheet 3:01 Pythagoras’ Theorem A, MS4·1 Measure each side of c this triangle. Work a out a2, b2 and c2. b Does c2 = a2 + b2?

For each of the triangles drawn below, measure the sides to the nearest millimetre and complete the table. a

b

a2

c

a

b2

c2

a 2 + b2

b

a

I see! c is the longest side of each triangle.

b c

a

c

b

c

d c

a

e b

c

b

a

c b

2

34

a

In which of the triangles in Question 1 does c2 = a2 + b2? What type of triangles are they?

NEW SIGNPOST MATHEMATICS 8

3

A square has been drawn on each of the sides of these right-angled triangles. Complete the tables for each triangle. Can you find a rule that links a, b and c?

a

b

A3

A3 5 3

A1

4

10

A1

6

A2

8

c A2

A3

13 5

a

A1

a

b

c

A2

A3

3

12 A1

9 units2 units2 units2

b A2

c a

b

c

a

b

c

A1

A2

A3

A1

A2

A3

units2 units2 units2

units2 units2 units2

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

35

E

4

D

F

3c

m

C

A sequence of triangles has been formed with a common base AB by gradually increasing the size of ∠B while keeping the side BC constant in length. The vertex C moves along a circle to the positions D, E and F.

4 cm

A

B

a Complete the table for the triangles named. Triangle

AB2

(right side)2

(left side)2

∆ABC ∆ABD ∆ABE ∆ABF b For what triangle does AB2 + (right side)2 = (left side)2? c What is the size of ∠B for the triangle in b?

inve

■ While answering the questions above you should have discovered, like the Pythagoreans, the relationship between the sides of a right-angled triangle.

stigation

Investigation 3:01B | Pythagoras’ Theorem Construct two identical squares as shown in the diagram. Use any value you like for a and b a c c (eg a = 6 cm, b = 8 cm). The coloured squares b b are built on the three sides of a right-angled triangle. a Is the area of square I equal to the area of a b a square II? a b b a b Do the triangles in square I take up the same I II areas as the triangles in square II? If the triangles are removed from squares I and II, are the remaining parts equal in area? If the triangles are removed from square I, what is the area of the two remaining squares? If the triangles are removed from square II, what is the area of the remaining square? Does c2 = a2 + b2? a

3:01B

b

a

c d e f

b

a

b

Complete: In any right-angled triangle the square on the . . . . . . . . . side is equal to the . . . . . . . . . of the squares on the other two . . . . . . . . .

36

NEW SIGNPOST MATHEMATICS 8

c

b

a

Pythagoras’ Theorem • In any right-angled triangle the square of the longest side is equal to the sum of the squares of the two shorter sides. For the triangle shown, the theorem can be written as: c2 = a2 + b2 Converse • If a triangle obeys the equation above, it must be right-angled.

Class activity The Pythagorean Theorem can also be illustrated by the following geometric method. Construct a right-angled triangle and construct squares on each side of the triangle as shown in the diagram. We have seen that for all right-angled triangles c 2 = a2 + b2 . Since the areas of the squares constructed on the sides are a2, b2 and c2 respectively, the theorem can be restated as:

c

b

a

The square on the hypotenuse is equal to the sum of the squares on the two smaller sides.

The larger square can be made from the two smaller squares by dissecting the middle square and rearranging these pieces with the small square. The dissection is shown in the following diagram. Copy the dissection and find how the pieces can be arranged to form the large square. I hope you’re good at jigsaw puzzles. B

C P

A

D

■ • P is the centre of the square. • AB is drawn through P, parallel to the hypotenuse. • CD is drawn through P, perpendicular to the hypotenuse.

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

37

3:02 | Using Pythagoras’ Theorem:

Outcome MS4·1

Calculating the Hypotenuse

pr

quiz ep

3:02

For the triangles given in 1 to 3, select the correct statements from A, B or C. 1 A 52 + 122 = 132 2 A 3+3=x 3 A 82 = 62 + x2 B (5 + 12)2 = 132 B 32 + 32 = x2 B x2 = 62 + 82 C 132 + 52 = 122 C (3 + 3)2 = x2 C x=6+8

5

13

6

x

8

3 x

12 3

5 If 42 = 16, then

4 Write Pythagoras’ Theorem for the triangle given.

6 If 252 = 625, then

n

7 If

625 = . . . .?

25 = 5 , then 52 = . . . .?

8 32 + 42 = . . . .?

m

c

16 = . . . .?

9

36 = . . . .?

10 Can 2, 3 and 4 be the side lengths of a right-angled triangle? • Pythagoras’ Theorem is used to calculate the lengths of the sides of right-angled triangles. To calculate one side, the other two sides must be known.

worked examples Calculate the length of the hypotenuse in each of the following triangles. 1 2 5 cm x cm

5 cm

3 cm a cm

12 cm

Solutions 1 x2 = 52 + 122 = 25 + 144 = 169 x = 169 = 13 Hypotenuse is 13 cm.

38

NEW SIGNPOST MATHEMATICS 8

2 a2 = 32 + 52 = 9 + 25 = 34 a = 34 a ⯐ 5·8 (to 1 dec. pl.) Hypotenuse is 5·8 cm.

■ Important Notice Numbers like 34 are called surds. Surds do not have exact decimal equivalents, so we use decimals to approximate their value.

Numbers without exact decimal equivalents? Why that‘s ABSURD!

Exercise 3:02A (No calculator is needed for this exercise)

15 cm

15 cm

9m

am

m

hc m

cc

8 cm

Use Pythagoras’ Theorem to write an equation that shows how the sides of each triangle are related to each other. a b c d 10 cm 8 cm

1

m

pc

12 m 10 cm

3

4

5

Square

9

16 25

6

7

8

9

Use the table in Question 2 to find: a 25 b 100 c 169

10 12 13 15 16 17 18 20 24 25 26 30

d

e

225

f

900

g

625

289

Find the length of the hypotenuse in each of the following. (Use the table in Question 2.) a b c d 6 cm 24 cm x yc

a cm

m

10 m

hm

8 cm

4

Number

7 cm

3

Complete the following table.

3m

2

m

24 m

4m

Find the length of the hypotenuse in each of the following. (Use the table below to evaluate the surds correct to 1 decimal place.) 10

13

17

18

20

26

29

32

34

2·2

2·8

3·2

3·6

4·1

4·2

4·5

5·1

5·4

5·7

5·8

b

a cm

d

m

m xc

b

5 cm

c

2m 2m

1 cm

a

8

2 cm

f 3 cm

g

2m 1m

cm

3m 2m

e

tm

hm

Approximate value

5

4 cm

Square root

p

5

4m

1m

Lm

3 cm

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

39

6

Use Pythagoras’ Theorem and the tables in Questions 2 and 5 to find the length of the diagonal and the perimeter of each square or rectangle. a b c 8 cm

4 cm 15 cm

3m

3m 4 cm

Use Pythagoras’ Theorem to find the value of pronumerals in each of the following. (You will need to use the tables in Questions 2 and 5.) a b c 3 cm

7

bm

am 13 m

y cm

6m

15 m

5m

4 cm

am

8m

15 m

8

Find the length of the interval AB in each of the following. y y a b A 4 B

3

2 1

9

40

B 2 3 4x

A –3 –2 –1

a The foot of a ladder is 1 m from a vertical wall. If the ladder reaches 5 m up the wall, find the length of the ladder, to the nearest centimetre. b Two roads meet at right angles. One person walks 6 km along one road while another person walks 8 km along the other road. How far apart are they? c A radio mast is supported by three wires. Each wire is attached to the mast 5 m from the bottom. The wires meet the ground at a point 2 m from the bottom of the mast. Find the length of each wire to the nearest tenth of a metre.

NEW SIGNPOST MATHEMATICS 8

y 5 B 4

2 1

c

2 x

A 2

4

6

Always draw a picture of the problem.

8 10 12 x

Exercise 3:02B (Calculators are needed) 1

2

Use the calculator to simplify: a 162 b 2·52 d 289 e 1681 g 96 (to 1 dec. pl.)

Foundation Worksheet 3:02

c 6·82 f 33·64 h 148 (to 2 dec. pl.)

Pythagoras’ Theorem B, MS4·1 1 Here c2 = 22 + 32 c = 13 2 ∴ c = 13 3 =.....

Use Pythagoras’ Theorem to calculate the length of the hypotenuse in each of the following. a b c d 16 cm 1·6 m 2 6

m

m

6m

cc

3·

cm

12 cm

cm

1·2 m

·7

m

cm

1·1 m

a

dm

m

c cm

4·5 m

12 cm

2 cm

m

bc

18 cm

3m

e

12 cm

Find the value of the pronumeral in each of the following, correct to 1 decimal place. a b c d 3·5 m 3m

3

f

3m

g 4

A square root?

80 cm

m

fc

gm

m

em

10 m

50 cm

m

xc

15 cm 6 cm

8 cm

16 m

8 cm

8m

m

5 cm

dc

12 cm

a

cm

9 cm

Find the value of the pronumeral in each of the following. Give answers correct to one decimal place where necessary. 20 m a b c d 6 cm

4

ym

e Find the perimeter of the kite in c. CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

41

inve

stigation

Investigation 3:02 | Pythagorean triads 3:02

A Pythagorean triad is a set of three numbers that obey Pythagoras’ Theorem. If 52 + 122 = 132, then {5, 12, 13} is a Pythagorean triad. 1 Use algebra to show that 3x, 4x, 5x is a Ah-ha! This means Pythagorean triad. that if {3, 4, 5} Use this result to complete the table below. a

b

c

3 6 9 ... ... 3·6

4 8 12 16 36 ...

5 ... ... 20 45 6

{3

is a triad, then 2, 4 2, 5 is also a triad.

2 This table shows a special set of Pythagorean triads. a Can you complete the next three lines of the table? b The numbers in the table can be found using the formula 2n + 1, 2n2 + 2n, 2n2 + 2n + 1. Use this formula to find the triads starting with: i 27 ii 51 iii 101

2}

a

b

c

3 5 7 9 11 13 15

4 12 24 40 ... ... ...

5 13 25 41 ... ... ...

There are many Pythagorean triads but often they are hard to figure out. Of course, you can make more triads by simply multiplying an existing one by a constant, such as: {3, 4, 5} × 2 ⇒ {6, 8, 10} The formula below, however, turns out triads by substituting odd numbers, starting at 3, for n. n2 ± 1 ------------■ 2 ‘±’ means ‘plus or minus’

Formula: n, n2 − 1, n2 + 1. 32 ± 1 If we put in 3 we get --------------2 ∴ the triad is {3, 4, 5}.

11 2 ± 1 If we put in 11 we get -----------------2 ∴ the triad is {11, 60, 61}.

11 2 + 1 ------------------ = 61 2 11 2 – 1 ----------------- = 60 2

32 + 1 --------------- = 5 2 32 – 1 -------------- = 4 2

■ Check! 112 + 602

= 121 + 3600 = 3721 Now 612 = 3721 So 112 + 602 = 612

3 Substitute further odd numbers for n and turn out some more triads. Check that they actually work by testing them with Pythagoras’ Theorem. 4 Make a list of Pythagorean triads that are made up of three numbers which are all less than 100.

42

NEW SIGNPOST MATHEMATICS 8

3:03 | Using Pythagoras’ Theorem:

Outcome MS4·1

Calculating One of the Short Sides

Pythagoras’ Theorem can also be used to calculate the length of either short side. The examples below show how this is done.

worked examples Find the value of x in each of the following. 1 2 m 13 c

xm

x cm

4m

8m

12 cm

Solutions 1 132 = 122 + x2 169 = 144 + x2 25 = x2 ∴ x = 25 =5

2 82 = x2 + 42 64 = x2 + 16 48 = x2 48 = x x = 48 ⯐ 6·93 m (correct to 2 dec. pl.)

If you want the good mail, always start the problems with: (long side)2 = (short side)2 (middle side)2.

Exercise 3:03A (No calculator is needed for this exercise)

13 m

cm

4 cm

15

5m 2

25 m

15

xm

cm

b cm

5

yc m

Use Pythagoras’ Theorem to write an equation that shows how the sides of each triangle are related to each other. a b c d 12 cm a cm

1

m

Use the following table to answer the questions below. Number

3

4

Square

9

16 25 36 49 64 81 100 144 169 225 256 289 324 400 576 625 676 900

a

9

b

5

6

144

7

8

c

9

400

10 12 13 15 16 17 18 20 24 25 26 30

d

324

e

64

f

81

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

43

Complete the working for each of the following. Use the table from Question 2 in all exercises. a 132 = 122 + x2 b 302 = 242 + a2 24 cm cm 169 = 144 + x2 900 = 576 + a2 13 2 x =........ a2 = . . . . . . . . cm 30 12 cm x= ........ a = ........ x=........ a=........

y cm

cm

d

8

pm m

262 = y2 + 102 . . . . = y2 + . . . . y2 = . . . . y=....

c

26

a cm

x cm

3

102 = p2 + 82 . . . . = p2 + . . . . p2 = . . . . p=....

m

10 m

10 cm

4

Using the setting out shown in Question 3, find the values of the pronumerals in Question 1.

5

a The hypotenuse of a right-angled triangle is 20 cm long. If one of the short sides is 12 cm long, what is the length of the other side? b The longest side of a right-angled triangle is 17 cm. If one of the other sides is 8 cm in length, find the length of the third side. c The two longer sides of a right-angled triangle are 26 cm and 24 cm in length. How long is the shortest side? d In a right-angled triangle the longest side is 15 cm long. The shortest side is 9 cm long. How long is the third side?

m

am

b cm

4m

m 13 c

5m

c 10

y cm

6 cm

d 10

cm

2 cm

Find the value of the pronumeral in each of the following. a b c

8 cm

6

The hypotenuse is the longest side.

h cm 5 cm

e Find the perimeter of the rectangle in b.

44

NEW SIGNPOST MATHEMATICS 8

fun spo

t

Fun Spot 3:03 | What happens when ducks fly upside down? Answer each question and put the letter for that question in the box above the correct answer.

3:03

The Pythagorean triad {3, 4, 5} has been multiplied by a constant to produce these triads. Find the missing number in each triad.

Exercise 3:03B (Calculators are needed)

Foundation Worksheet 3:03 Pythagoras’ Theorem, MS4·1 1 b2 + 52 = 72 5 b2 + 25 = 49 ∴ b2 = . . . . b 7 ∴ b= .... =.... 2 a 2 a + 52 = 132 a2 + . . . . = . . . . 5 ∴ a2 = . . . . 13

Find the value of the pronumeral in each of the following. a b 3·6 m 3· 3m

3·9 m

c

d

28 m

cm

a cm

50

y cm

pm

4m

xm

1

2·4

44

5

3·5

9

70

1·5

6·6

16

T 12, . . . ., 20 E . . . ., 2, 2·5 U 27, . . . ., 45

. . . ., 12, 15 1·8, . . . ., 3 42, 56, . . . . 3·75, . . . ., 6·25

36

A P Y K

6, 8, . . . . 2·1, 2·8, . . . . 33, . . . ., 55 . . . ., 8·8, 11

10

Q C U H

35 m

30 cm

am

8m

4·5 m

8 cm

3·8 m

xm

m

9c

y cm

Find the value of the pronumeral in each of the following, correct to 1 decimal place. a b c d 4m 3·2 m x cm

2

3m CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

45

m

70 m

m

2m

6·5

ym

2·5

30 m

Find the value of the pronumeral in each of the following. Give answers correct to 1 decimal place where necessary. a b c hm Lm

3

104 m 2·5 m

a A ladder that is 13 m long is placed so that its foot is 3 m from the base of a vertical wall. How far does the ladder reach up the wall?

13 m

4

hm

3m

hm

b A piece of wire 20 m long has one end attached to the top of a vertical building. The other end is fixed to the ground at a distance of 5 m from the building. Find the height of the building, correct to 2 decimal places.

20 m

5m

c A ship is 15 nautical miles (nm) from port. After sailing 5 nm due south, it is directly east of the port. How far is the ship then from the port?

Port

5 nm

d nm

xm

2·4 m

Lm

d Find the value of h. 5m

6m

3·6 m

hm

13 m

15 m

hm

5m

c Find the values of h and L. m

b Calculate x and y.

hm

a The diagram shows a cone with a radius of 3·6 m and a slant height of 6 m. Calculate the height, h, of the cone.

4

5

15 nm

9·6 m 7·68 m

ym 6

46

a The perimeter of a square is 16 cm. Find the length of its diagonal, correct to 1 decimal place. b A rectangle’s length is twice its breadth. Find the length of its diagonal (to 1 dec. pl.) if its perimeter is 90 cm.

NEW SIGNPOST MATHEMATICS 8

3:04 | Miscellaneous Exercises

Outcome MS4·1

Involving Pythagoras’ Theorem

The following exercise consists of a mixture of questions in which Pythagoras’ Theorem will need to be used. In some questions you will be asked to calculate the length of the hypotenuse, while in others you will need to calculate the length of the shorter side.

Exercise 3:04 (Calculators are needed) Find the value of the pronumeral in each of the following. a b c 12·8 cm

d

17

cm

xm

9m

1

9·6 cm

pc

a cm

4·5 m

m

bm

6m

12 m

15 c

e

m

8 cm x cm

f

g 50 m ym

3·6 m

25 cm

tm

48 m 4·8 m

a Fiona measured the two shorter sides of a right-angled triangle and found them to be 16 cm and 30 cm. She then measured the longest side. What should its length have been? b Gino had an extension ladder that was 12 m long. If he placed the ladder against a wall so that the foot was 2 m from the building, how far up the wall would it reach, to the nearest centimetre? c Chiang and Su-lin start at A. B Chiang Chiang walks towards B while Su-lin walks towards C. How far apart are they when Chiang has walked 4 km and Su-lin Su-lin 3·6 km? Answer correct 3·6 km A C to 1 decimal place. d Larry Bagnell needs to repair the guttering on his house. From the diagram calculate the length of the smallest ladder he needs to hire, if the ladder has to extend at least 1 m past the gutter. e Angelika is building a barbecue on a rectangular concrete slab. The slab is to be 2·4 m long and 1·8 m wide. What should the length of the diagonal be?

Draw a diagram if one is not given.

7·2 m

1m

4 km

2

1·8 m

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

47

BILLABONG RD 3000 m

1600 m

Jumbuck Ave

f

Billabong Rd and Jumbuck Ave meet at right angles as shown in the diagram. Peter Perfect wants to go from A to B. What distance does he save by taking the shortcut down Nobb’s Lane?

B

s bb’

e Lan

No

A

g Is a triangle with sides 3·6 m, 10·5 m and 11·1 m in length right-angled? h

The diagram shows a gate. Calculate the length of timber that would need to be bought to make the brace.

ce 1·2 m bra 2·1 m

i

Two roads intersect at right angles. Two drivers, Bill and Bev, leave the intersection at the same time. Bill drives down one road at an average speed of 60 km/h while Bev drives down the other road at an average speed of 80 km/h. Find how far each person has driven in half an hour and how far apart they are at that time.

j

Part of a roof truss is shown in the diagram. Calculate the length of the rafter to the nearest ten millimetres.

1485 mm

raft e

r le

8100 mm

ngt

h

900

mm

2400 mm

k Jill and John decide to cover their bathroom wall with cedar boards as shown in the diagram. What is the longest length board that needs to be bought so that there would be no joins?

3300 mm

l

Two posts are 6 m tall and 8 m tall. If these posts are 10 m apart on level ground and a string is stretched from the top of one post to the top of the other, find the length of the string, to the nearest centimetre.

8m

6m 10 m

36 cm

m

x cm

m

c 45

7c

1·4

Use Pythagoras’ Theorem to find the values of the pronumerals in each of the following. a b c m ■ Start with 4·8 yc m m c xm a triangle in x which two sides are 4·2 cm 10·5 cm y cm 48 cm 4m known. m

3

ym

48

NEW SIGNPOST MATHEMATICS 8

4

a Find the perpendicular distance from the corner of the triangle to the opposite side if the side is 25 cm long.

b Find the length of the diagonal if the length of the window is 48 cm and its breadth is 35 cm.

c The diameter of the cone in this hat is 18 cm. Its slant height, s cm, is 21 cm. What is the height of the hat?

d The legs of this ladder are of equal length and are 1 metre apart. The slant height of the ladder is 2·4 metres. How high above the ground does the ladder reach?

b

m

c 15

x cm 12 cm

The figure is a triangular prism, Find the value of x, correct to 2 decimal places.

y cm

5 cm

a 5 cm

5

6c

m

m

8c

The figure is a rectangular prism. Find the value of y, correct to 2 decimal places.

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

49

inve

stigation

Investigation 3:04 | Pythagoras and speed Wendy usually walks to the railway station along roads shown in the diagram below. 3:04

Paddock C

Paddock B 1200 m Housing (No shortcuts) 600 m

Paddock A

350 m

STATION

500 m Wendy’s house

0

40

12

60

14

30

0 15

20

0

13

0

50

0

NEW SIGNPOST MATHEMATICS 8

110

80 7 0

10

50

100 90

170 180

Maths terms 3

C

100

20

B

80

160

A

10 01

30

perpendicular • Meeting at right angles. eg AB is perpendicular to BC, ie AB ⊥ BC

12

0

Pythagoras’ theorem • In any right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

0 13

15

hypotenuse • The longest side of a right-angled triangle.

70

60 50

40

protractor • A geometrical instrument used to measure angles.

0 14

diagonal • A straight line that crosses a polygon to join two of its vertices.

10

3

Maths terms 3

0

rm s te s

180 170 1 60

math

Recently, after sleeping in, she needed to get to the station in a hurry and decided to take shortcuts through the paddocks. Wendy can run at 10 km/h and walk at 5 km/h. If she has only 18 minutes before the train leaves, should she: i walk (arriving on time and unflustered)? ii run (because she will be late otherwise)? iii walk part of the way and run the rest? iv change her plans (as she just can’t make it)? Explain your answer in sentences.

c

a

b

c2

= a 2 + b2

es tic t t

diagnos

Diagnostic Test 3: | Investigating Pythagoras’ Theorem • Each section of the test has similar items that test a certain type of example. • Failure in more than one item will identify an area of weakness. • Each weakness should be treated by going back to the section listed. These questions can be used to assess part of outcome NS4·1. Section 3:01

1 State whether the given triangle is right-angled. a b c

2 In each of the following, list the sides in order of length from shortest to longest. a b c m

c

b

n

l

n

3 Use Pythagoras’ Theorem to write an equation that shows how the sides of each triangle are related to each other. a b c 8m m

6m

m

3:02

4·2

6m

6

5·

Lm

cm

3:01

p

a

m

3

xm

3m

5 Use Pythagoras’ Theorem to write an equation that shows how the sides of each triangle are related to each other. a b c am 4m 20 m

3:03

m

b

2

8m

m

3·

17

a cm

3:02

m

4 Find the length of the unknown side in each of the triangles in Question 3.

25 m

3:04

34 m

m yc

am

4·2 m

7 Find the length of the unknown side in each of the triangles below. a b c 36 cm 15 cm

3:03

30 m

6 Find the length of the unknown side in each of the triangles in Question 5.

5·6 m

xm

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

51

assi

men gn t

Chapter 3 | Revision Assignment 1

4

3A

C D

A

B

a Name solid A. b Name solid B. c Calculate the number of faces, edges and vertices in solid B. d Name the cross-sectional shape of the solid formed by placing solid A on top of solid B. 2 The nets of three solids are given below. Name the solid formed from each net. a

b

A

B

a What type of angle is ∠ABC? b What type of angle is ∠BAC? c By estimating, choose the angle that is closest in size to a right angle. d Name an obtuse-angled isosceles triangle. 5 Change these Roman numerals into our own numerals. a XLVIII b CIX c MDLXV d MCMXCIII 6 Complete the following conversions. a 5 m = mm b 2·5 m = cm c 5·2 g = mg d 560 kg = t e 3300 mL = L 7 A property is rectangular in shape. It is 110 m long and 22 m wide. Calculate the cost of fencing the block if fencing is $8.40 a metre. 8 Write each time below as a digital time. a 20 past 11 in the morning b a quarter to 8 in the evening 9 Calculate the areas of the figures shown. a 2·5 m

c

8m

b

42 mm 68 mm

c 2·1 m

6·5 m

3 Classify the following angles as acute, obtuse or reflex. a 78° b 155° c 98° d 185° e 278° 4m

52

NEW SIGNPOST MATHEMATICS 8

10 Calculate the volume of the figures shown. a b

c

1·2 cm

3 cm

4·5

1 cm

cm

3 cm 5 cm

3 cm

assignm

1 Eight dots are drawn on a page. If no three dots are in the same straight line, how many intervals can be drawn that join any two dots? 2 Fred Shipshape built a sailing ship to mark Australia’s 200th anniversary of European settlement. He wanted to use two words in naming it, but both words could not be the same. How many names were possible if the first name had to be either Endeavour, Australia, National, Young or Shining, and the second name had to be either Endeavour, Australia, Pride, Supreme, Courage or Star?

ent

Chapter 3 | Working Mathematically 3 Five times Kate’s age plus twice Sarah’s age is equal to 25. List the possible ages for these two girls.

3B

4 On a digital clock, how many times between 12 noon and 12 midnight will the readout be a palindromic number? (See Appendix A:02D, for a definition of palindromic numbers.) 5 How many integers are less than 5 units from the number two? (An integer is any whole number, positive, negative or zero.) 6 Drinks cost $1.20 each and ice-creams 80c each. The drinks and ice-creams for our group cost $12.40. If 12 items were purchased, how many drinks did we buy?

1 Number properties 2 Language and symbols 3 Special sets of numbers

3:04 Pythagorean triads

Pythagoras’ Theorem

CHAPTER 3 INVESTIGATING PYTHAGORAS’ THEOREM

53