PYTHAGORAS’ THEOREM Pythagoras’ Theorem {, ## # = = = += + =-= - = ` ` {,-2 ’ M ’ M A * A...

Pyth

agor

as’ T

heor

em PYTHAGORAS’ THEOREMPYTHAGORAS’ THEOREMPYTHAGORAS’ THEOREM

www.mathletics.com.au

7ISERIES TOPIC

1Pythagoras’ Theorem Solutions

Mathletics Passport © 3P Learning

How does it work? Pythagoras’ TheoremSolutions

2

M

N

L

a b

Hypotenuse is side: y

Hypotenuse is side: DF

Hypotenuse is side: PQ

Hypotenuse is side: k

a b

c d

y

Q

RP

D

F

E

x

z

k

jl

Hypotenuse is side: a Hypotenuse is side: MN

Name the hypotenuse for each of these badly drawn triangles.

questions

Right-angled triangles

ca

b

1

2 Pythagoras’ Theorem Solutions


7ISERIES TOPIC


1

2

Area 1 = 5 units # 5 units = 25 units2



Area 1 + Area 2 = 25 units2 + 144 units2




Area 1 + Area 2 = 36 units2 + 64 units2

1

3

2

13 units

12 units

5 units

3

2

110 units

6 units

8 units

Page 5 questions

Squares and right-angled triangles

= 169 units2

= Area 3

= 100 units2

= Area 3

7ISERIES TOPIC




15

12

9

a b

c d

Not right-angled

e f

Right-angled

25

2014

Not right-angledRight-angled

3.4

9.6

7.1


1.2

3.5

3.7


21

29

20


25

7

24


Page 7 questions

Pythagoras’ Theorem for right-angled triangles

1

9 12 81 144 15 225

225

2 2 2+ = + =

=

14 20 196 400 25 625

596

2 2 2+ = + =

=

7.1 3.4 50.41 11.56 9.6 92.16

.61 97

2 2 2+ = + =

=

1.2 3.5 1.44 12.25 3.7 13.69

.13 69

2 2 2+ = + =

=

20 21 400 441 29 841

841

2 2 2+ = + =

=

24 7 576 49 25 625

625

2 2 2+ = + =

=



7ISERIES TOPIC


16

20

25 20

A

B

J

12

KI

J

1010.5

14.5H

15AC

48

20

52

H

G

K

The right-angled triangles are: ΔAJK , ΔHIJ , ΔGHK

Page 8 questions


2

12 16 20

400 400

2 2 2+ =

=

. .

. .

10 10 5 14 5

210 25 210 25

2 2 2+ =

=

20 15 24

625 576

2 2 2

!

+ =

N

M

L

21

29

48

29 21 48

1282 2304

2 2 2

!

+ =

48 20 52

2704 2704

2 2 2+ =

=

7ISERIES TOPIC




3 Earn an awesome passport with this one! Name all the right-angled triangles in this image and markwhere the right-angles are with the correct symbol.

65R S

522436

155

1612

P QU

T

153280

V

The right-angled triangles are:

ΔPUV

ΔQRU

ΔRSU

ΔSTU

15 36 39

1521 1521

2 2 2+ =

=

39 52 65

4225 4225

2 2 2+ =

=

24 52 3280

3280 3280

2 2 2+ =

=

12 16 20

400 400

2 2 2+ =

=

Page 9 questions


Assuming the scale of the page is the same as the original print, the measurements should be as follows:NOTE: if not the same scale, the same relationship between your measurements should work.

Page 8 questions


202545 mm 360060 mm 562575 mm 2025 + 3600 = 5625

490070 mm 57624 mm 547674 mm 4900 + 576 = 5476

22515 mm 129636 mm 152139 mm 225 + 1296 = 1521

1

2

3

4

5

6

129636 mm 592977 mm 722585 mm 1296 + 5929 = 7225

819 mm 160040 mm 168141 mm 81 + 1600 = 1681

160040 mm 176442 mm 336458 mm 1600 + 1764 = 3364

a2a b2b c2c a2 + b2



7ISERIES TOPIC

Where does it work? Pythagoras’ TheoremSolutions

1 a 6 8

36 64

100

c

c

c

c

c

100

10

2 2 2

2

2

= +

= +

=

=

=

`

`

`

`

b 8 15

64 225

289

g

g

g

g

g

289

17

2 2 2

2

2

= +

= +

=

=

=

`

`

`

`

2 a b

Page 11 questions

Calculating the length of the hypotenuse

5 12

25 144

169

c

c

c

c

c

169

13

2 2 2

2

2

= +

= +

=

=

=

`

`

`

`

1.2 1.6

1.44 2.56

4

d

d

d

d

d

4

2

2 2 2

2

2

= +

= +

=

=

=

`

`

`

`

Page 12 questions


a 1.1 6.0

1.21 36

37.21

.

h

h

h

h 37 21

2 2 2

2

2

= +

= +

=

=

`

`

`

b 12 35

144 1225

1389

n

n

n

n 1389

2 2 2

2

2

= +

= +

=

=

`

`

`

a bunits units

units units

units

units

units

( ) ( )

. ...

.

c

c

c

c

c

c

10 9

100 81

181

181

13 45362405

13 45

2 2 2

2 2 2

2 2

.

= +

= +

=

=

=

`

`

`

`

`

`

`

`

4

3

units units

units units

units

units

units

( . ) ( . )

. .

.

.

. ...

.

p

p

p

p

p

p

5 9 3 4

34 81 11 56

46 37

46 37

6 809552114

6 81

2 2 2

2 2 2 2

2 2

.

= +

= +

=

=

=

units to 2 decimal placesunits to 2 decimal places

in exact square root formin exact square root form

``

7ISERIES TOPIC




a b

2

Page 13 questions


5

m

40 198

1600 39204

40804

d

d

d

d

d

40804

202

2 2 2

2

2

= +

= +

=

=

=

Stage 1

m

39 252

1521 63504

65025

d

d

d

d

d

65025

255

2 2 2

2

2

= +

= +

=

=

=

Stage 2

m

m

. ...

.

d

d

d

d

d

d

36 360

1296 129600

130896

130896

361 7955224

361 8

2 2 2

2

2

.

= +

= +

=

=

=

Stage 3

`

`

`

`

`

`

`

`

`

`

`

`

` The total length of the 3 stage flight path 202 255 361.8. + + m

m818.8.

1

Page 15 questions

Calculating the length of a short side

26 24

676 576

100

a

a

a

a

a

100

10

2 2 2

2

2

= -

= -

=

=

=

8.5 1.3

72.25 1.69

70.56

.

.

b

b

b

b

b

70 56

8 4

2 2 2

2

2

= -

= -

=

=

=

`

`

`

`

`

`

`

`

a b70 56

4900 3136

1764

j

j

j

j

j

1764

42

2 2 2

2

2

= -

= -

=

=

=

units units

units units

units

units

units

(18.1 ) (18 )

327.61 324

3.61

.

.

b

b

b

b

b

3 61

1 9

2 2 2

2 2 2

2

= -

= -

=

=

=

`

`

`

`

`

`

`

to 2 decimal places

`

`



7ISERIES TOPIC


a b

. .

. .

.

.

.

.

x

x

x

x

x

x

41 08 23 42

1687 5664 548 4968

1139 07

1139 07

33 7501

33 8

2 2 2

2

2

.

= -

= -

=

=

=

`

`

`

`

3

Page 16 questions

Calculating the length of a short side

17 11

289 121

168

b

b

b

b 168

2 2 2

2

2

= -

= -

=

=

`

`

`

14.25 11.75

203.0625 138.0625

65

w

w

w

w 65

2 2 2

2

2

= +

= +

=

=

`

`

`

4 . .

. .

.

.

. ...

.

y

y

y

y

y

y

13 8 8 3

190 44 68 89

121 55

121 55

11 02497166

11 0

2 2 2

2

2

.

= -

= -

=

=

=

`

`

`

`

to 1 decimal point

a b

` to 1 decimal point

in exact square root formin exact square root form

`

7ISERIES TOPIC




= triangle

15

20

545

544

42

42.1

41.9

53.2

32

68

20

14.16

30

67

g

h

e

d

b

a

c

14.12

2.9

73.4

25

67.7

33

60

1 2 3 4 5 6

Page 17 questions

Combination of hypotenuse and short side calculations

The special name given a right-angled triangle which is exactly one half of an equilateral triangle:

H E M I E Q

MI

E

H

Q

E

2

1

3

4

5

6



7ISERIES TOPIC


3

(ii) To avoid the swamp, Mila walked 3.9 km + 1.7 km = 5.6 km

x

12 m13 m

Start1.7 km

Finish

3.9 km

2

Page 19 questions

Applications of Pythagoras’ Theorem

1

cm cm

cm cm

cm

cm

cm

(42 ) (34 )

1764 1156

2920

54.03702434

cut

cut

cut

cut

cut

cut

2920

54

2 2 2

2 2 2

2 2

.

= +

= +

=

=

=

`

`

`

`

`

m m

m m

m

m

m

(13 ) (12 )

169 144

25

5

x

x

x

x

x

25

2 2 2

2 2 2

2 2

= -

= -

=

=

=

`

`

`

`

to nearest whole cm

km km

km km

km

km

km

km

(1.7 ) (3.9 )

2.89 15.21

18.1

.

4.254409477 ...

4.25

d

d

d

d

d

d

18 1

2 2 2

2 2 2

2 2

.

= +

= +

=

=

=

`

`

`

`

` to 2 decimal points

(i)

` Mila walked a further 5.6 km - 4.25 km . 1.35 km

d

42 cm

34 cm cut

7ISERIES TOPIC




3.3 m

2.6 m17 m

(i)

(ii)

C

18 m

54.4 m

A

B

Diagram not drawn to scale.

base

4

Page 20 questions


m m

m m

m

m

m

(17 ) (2.6 )

289 6.76

282.24

.

16.8

base

base

base

base

base

282 24

2 2 2

2 2 2

2 2

= -

= -

=

=

=

`

`

`

`

m m

m

m

( . . )

43.68 2

.

16 8 2 6 2

21 84

2

2

# '

'

=

=

=

`

`

Area = (base # height) ' 2

Area

Area

Area`

5

AB m m

AB m m

AB m

AB m

AB m

(18 ) (3.3 )

324 10.89

334.89

.

18.3

334 89

2 2 2

2 2 2

2 2

= +

= +

=

=

=

`

`

`

`

BC m m

BC m m

BC m

BC m

BC m

(54.4 ) (3.3 )

2959.36 10.89

2970.25

.

54.5

2970 25

2 2 2

2 2 2

2 2

= +

= +

=

=

=

`

`

`

`

Distance AC AB BC

m m

m

18.3 54.5

72.8

= +

= +

=

Distance around wall = 79m

Shortest path



7ISERIES TOPIC


6 172 cm

120 cm

137 cm

y

Page 20 questions


cm cm cm

cm cm

cm cm

cm

cm

cm

(172 137 ) (120 )

(35 ) (120 )

1225 14400

15625

125

y

y

y

y

y

y

15625

2 2 2

2 2 2

2 2 2

2 2

= - +

= +

= +

=

=

=

`

`

`

`

(i)

(ii) Perimeter of the trapezium cm cm cm cm

cm

172 120 137 125

554

= + + +

=

WY YZ WZ

WY

WY 63

65 16

4225 256

3969

3969

2 2 2

2 2

= -

= -

= -

=

=

=65

34

16

Y

X

W

Page 21 questions


7

WX XZ WZ

WX

WX 30

34 16

1156 256

900

900

2 2 2

2 2

= -

= -

= -

=

=

=

XY WY WX

63 30

33

= -

= -

=

` `

7ISERIES TOPIC




A

C

B

6 m

18.5 m

9.5 m

D

8 Calculate the length of the cable support BD on the crane picture below if CD = 9.5 m, AB = 6 m and BC = 18.5 m

AC BC AB

AC

AC

18.5 6

.

.

.

.

342 25 36

306 25

306 25

17 5

2 2 2

2 2

= -

= -

= -

=

=

=

AD AC DC

. .17 5 9 5

8

= -

= -

=

`

BD AD AB

BD

BD

8 6

64 36

100

100

10

2 2 2

2 2

= +

= +

= +

=

=

=

`

Page 21 questions


m

m

m



7ISERIES TOPIC

What else can you do? Pythagoras’ TheoremSolutions

1

2 Show whether these sets of positive integers form a Pythagorean triad or not.

a , ,7 24 25" , b , ,14 48 50" , c , ,12 34 36" ,

d , ,15 36 39" , e , ,16 60 63" , f , ,12 30 31" ,

Yes No Yes No Yes No

Yes No Yes No Yes No

1220

16

35

1237

26

1024

941

40

psst! Note that they are written in order of size.

b

c

d

, ,12 16 20" , , ,10 24 26" ,

, ,12 35 37" , , ,9 40 41" ,

?

?

7 24 25

49 576 625

625 625

2 2 2+ =

+ =

=

?

?

14 48 50

196 2304 2500

2500 2500

2 2 2+ =

+ =

=

?

?

12 34 36

144 1156 1296

1300 1296

2 2 2

!

+ =

+ =

?

?

15 36 39

225 1296 1521

1521 1521

2 2 2+ =

+ =

=

?

?

16 60 63

256 3600 3969

3856 3969

2 2 2

!

+ =

+ =

?

?

12 30 31

144 900 961

1044 961

2 2 2

!

+ =

+ =

Page 23 questions

Pythagorean triads

a

7ISERIES TOPIC




2 (i) Find a Pythagorean triad in which p = 7 and p q2 2- is equal to 33

(ii) Find a Pythagorean triad in which q = 5 and p q2 2+ is equal to 61

2 1 32 2- = 2 2 1 4# # = 2 1 52 2

+ = { , , }3 4 5

3 1 82 2- = 2 13 6# # = 3 1 102 2

- = { , , }6 8 10

5 2 212 2- = 2 5 2 20# # = 5 2 292 2

+ = { , , }20 21 29

7 6 132 2- = 2 7 6 84# # = 7 6 852 2

+ = { , , }13 84 85

11 3 1122 2- = 2 11 3 66# # = 11 3 1302 2

+ = { , , }66 112 130

21 18 1172 2- = 2 2 11 8 756# # = 21 18 7562 2

+ = { , , }117 756 765

p2 - q2 p 2pq p2 + q2 Triadq

2 1

3 1

5 2

7 6

11 3

21 18

Page 25 questions

Euclid’s formula for Pythagorean triads

1

33

7 33

49 33

49 33

16

p q

q

q

q

q

q 4

2 2

2 2

2

2

2

- =

- =

- =

- =

=

=

2 2 7 4pq

56

# #=

=

7 4p q

65

2 2 2 2+ = +

=

Pythagorean triad is { 33 , 56 , 65 }

p q

p

p

p

p

p

61

5 61

25 61

61 25

36

6

2 2

2 2

2

2

2

- =

- =

- =

= -

=

=

pq2 2 6 5

60

# #=

=

6 5p q

11

2 2 2 2+ = -

=

Pythagorean triad is { 11 , 60 , 61 }

`

`



7ISERIES TOPIC


Find a group of three integers that includes the number 14 and forms a Pythagorean triad.

hint: Pythagorean triads can be made using positive integers only.

Formal explanation:

3

4

Page 26 questions

Pythagorean triads

{ , 2 , }p q pq p q2 2 2 2- +

pq

p q

p q

2 14

2 14

7

# #

#

=

=

=

p 7= ( )q p q1 2=and

7 1p q

50

2 2 2 2+ = +

=

7 1p q

48

2 2 2 2- = -

=

` `

Pythagorean triad is: { , , }14 48 50

{ , 2 , }p q pq p q2 2 2 2- +

small integer other small integer largest integer

Showing using chosen values 2p = and 1q = :

From hint, Pythagorean triads are made using positive integers. ie. positive whole numbers only.One of the smaller integers is found using p q2 2

-

If the value of p was smaller than the value of q, then the answer would be negative. So this could not be used because only positive whole numbers are allowed.

When 2p = and 1q = , p q 2 1 32 2 2 2- = - = (this is a positive integer and is allowed)

If we swap these around, so 2p = and 2q = p q 1 2 32 2 2 2

- = - = - (this is a negative integer and is not allowed)

This will always happen if the value of p is smaller than the value of q when using Euclid’s formula.

Negative numbers are not allowed because each integer represents the length of the side of a right-angled triangle. So a side length of -3 does not make sense.

7ISERIES TOPIC




Page 28 questions

Wheel of Theodorus

812

16

20

24

2

2

2 2

2

2

2

and so on



7ISERIES TOPIC

What else can you do? Pythagoras’ TheoremSolutionsJigsaw Puzzle Pythagoras’ TheoremSolutions

2

1

3

4

4

3

2

1

5

55

Page 31 questions

Squares and right-angled triangles: Jigsaw Puzzle

7ISERIES TOPIC



Pythagoras’ Theorem Notes



7ISERIES TOPIC

Pythagoras’ Theorem Notes

APPLICATIONS

OF

PYTHA

GORAS’

THEOR

EM

..../...

../20...

PYTHAGORAS’ THEOREM

APPLICATIONS OF

RIGHT-ANGLED

TRIANG

LES RIGHT-ANGLED

TRIANGLES

..../...../20...

*

AWESO

ME *

..../.....

/20...

* AWESOME *

SQUARES AND RIGHT- ANGLED TRIANGLES SQUARES AND RIGHT- AN

GLED

TRI

ANGL

ES

..../...../20...

EU

CLID’S FORMULA FOR PYTHAGOREAN

TRAIDS

..../...../20...

, ,p q pq p q

22

22

2-

+

",

*

PYTHAGORAS’ THEOREM Pythagoras’ Theorem {, ## # = = = += + =-= - = ` ` {,-2 ’ M ’ M A * A...

Documents

Transcript of PYTHAGORAS’ THEOREM Pythagoras’ Theorem {, ## # = = = += + =-= - = ` ` {,-2 ’ M ’ M A * A...