Pythagoras Mathaletics

PASSPORT

PASSPORT

Pyth

agor

as’ T

heor

em PYTHAGORAS’ THEOREMPYTHAGORAS’ THEOREMPYTHAGORAS’ THEOREM

www.mathletics.com.au

7ISERIES TOPIC

1Pythagoras’ Theorem

Mathletics Passport © 3P Learning

The numbers 3, 4, 5 have the following relationship:

3 4 5

9 16 25

2 2 2+ =

+ =

Find another group of three whole numbers that includes the number 14 and has the same relationship.psst! the other two numbers are somewhere between 45 and 55!

Work through the book for a great way to do this

Give this a go!

Fill in these spaces with any other interesting facts you can find about Pythagoras.

One of his most recognised discoveries was the relationship between the side lengths of all right-angled triangles.

In the world of Mathematics, Pythagoras is a legend. He lived from 580 BC – 500 BC.

2 Pythagoras’ Theorem


7ISERIES TOPIC

Pythagoras’ TheoremHow does it work?

Right-angled triangles

These special triangles all have a right-angle (angle of size 90o) as one of the internal angles.

For each of these right-angled triangles, name the hypotenuse and then draw in the right-angle.

Short side

Hypotenuse(Longest side)

Other short side

opposite

(i) (ii)

The hypotenuse is the longest side

` hypotenuse = side c

The right-angle is the angle opposite the hypotenuse

The hypotenuse is the longest side

` hypotenuse = side XZ

The hypotenuse is always opposite the right-angle

Z

Y

Xa

c

b

Z

Y

X

Sides are lower case and corners are CAPITALS

opposite

hypotenuse

The two shorter sides are always perpendicular to each other.

Perpendicular = 90o

opposite

hypotenuse

a

c

b

Right angle

7ISERIES TOPIC



How does it work? Your Turn Pythagoras’ Theorem


1 For each of these right-angled triangles, name the hypotenuse and draw the right-angle in the correct position.

2

M

NL

a b

Hypotenuse is side:

Hypotenuse is side:

Hypotenuse is side:

Hypotenuse is side:

a b

c d

y

Q

RP

D

F

E

xz

k

jl

ca

b

Hypotenuse is side: Hypotenuse is side:

Name the hypotenuse for each of these badly drawn triangles:

RIGHT-ANGLED

TRIANG

LES

RIGHT-ANGLED TRIANGLES..../...../20...



7ISERIES TOPIC

How does it work? Pythagoras’ Theorem

Squares and right-angled triangles

When squares are drawn using each side length of a right-angled triangle, something interesting happens.

For the triangle below:

(i) Use the side lengths in the triangle to create three squares.

(ii) Calculate the area of each square formed and write a relationship between them. Area

Area

Area

5 units4 units

3 units

5 units

4 units

3 units

3 units

4 units

5 units

2

1

3

1

2

3

4 4 4 16

3 3 3 9

5 5 5 25

2

2

2

#

#

#

= = =

= = =

= = =

Area of a square = (side length)2

Area 1 + Area 2 = Area 3

16 units2 + 9 units2 = 25 units2

7ISERIES TOPIC




Squares and right-angled triangles

Show that the relationship between the areas of the squares formed using each side length works for these right-angled triangles:

1

2

3

Area 1 =

Area 2 =

Area 3 =

5 units13 units

12 units

10 units

Try the jigsaw puzzle at the back of this booklet to see another way of showing this property.

6 units

8 units

Area 1 =

Area 2 =

Area 3 =

1

3

2

13 units

12 units

5 units

3

2

110 units

6 units

8 units

SQUARES AND RIGHT- ANGLED TRIANGLES SQUAR

ES AND

RIGHT

- ANGLED TRIANGLES

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



7ISERIES TOPIC

How does it work? Pythagoras’ Theorem

Pythagoras’ Theorem for right-angled triangles

The squares and right-angled triangles section showed that a relationship exists between the side lengths of right-angled triangles. This relationship is called Pythagoras’ Theorem.

Use Pythagoras’ Theorem to determine which of the following triangles are right-angled or not.

(i)

cother short sidehypotenuse

short side a

b

If the rule does not work, then it is not a right-angled triangle.

(short side)2 + (other short side)2 = (longest side)2 a2 + b2 = c2

always the hypotenuse

4 6 822 2+ =

16 36 64+ =

52 64!

0.8 1.5 1.72 2 2+ =

0.64 2.25 2.89+ =

2.89 2.89=

` not a right-angled triangle ` is a right-angled triangle

8

4

6

1.5

1.7

0.8

(ii)

Substitute lengths into Pythagoras’ Theorem

(short side)2 + (other short side)2 = (longest side)2

7ISERIES TOPIC





Use Pythagoras’ Theorem to calculate which of the following triangles are right-angled or not.1

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


PYTHAGORAS’ THEOREM

FOR RIGHT-ANGLED TRIANGLES FOR RIGHT-ANGLED TRIANGLES

PYTHAGORAS’ THEOR

EM

ca

b

22

2=

+

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



7ISERIES TOPIC



Name all the right-angled triangles pictured below and mark where the right-angle is with the correct symbol.

16

20

24 20

2

3

A

B

J

12

K I

J

10 10.5

14.5H

15AC N

M

L

21

29

48 48

20

52

H

G

K

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.

The right-angled triangles are:

The right-angled triangles are:

Remember, triangles are named by their vertices.

D

EF

= ΔDEF

Diagram not drawn to scale.

65R S

522436

155

1612

P QU

T

153280

V

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

* AWESOME *

*

AWESOM

E *

7ISERIES TOPIC




Pythagoras’ Theorem by measurement

For each of these right-angled triangles:

(i) Use a ruler to carefully measure the length of each side to the nearest whole millimetre. (ii) Use the measurements to complete the table at the bottom of the page.

ca

b

c a

b

c

a

b

c

a

b

c

a bc

a

b

12

3 4

6

5

PYTHAGORAS’THEOREM BY MEASUREMENT PYTH

AGOR

AS’T

HEOR

EM

B

Y MEASUREMENT

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

1

2

3

4

5

6

a2a b2b c2c a2 + b2



7ISERIES TOPIC

Pythagoras’ TheoremWhere does it work?

Calculating the length of the hypotenuse

We can use Pythagoras’ Theorem to calculate the length of the hypotenuse if the two shorter sides of the right-angled triangle are already known.

The order that we put the short side values into the formula does not matter.

Calculate the length of the hypotenuse for this right-angled triangle.

Hypotenuse2 = short side2 + other short side2

Let’s label the hypotenuse ‘c’

24

7c

c

c

c

7 24

49 576

625

2 2 2

2

2

= +

= +

=

c

c

625

25

=

=

or c 24 72 2 2= + — short side order does not matter

To calculate the length c, square root this value

Write the positive answer only because it’s a length

Calculate the length of the hypotenuse for this right-angled triangle accurate to 2 decimal places.


8.16 units

m

. .

. .

.

m

m

m

8 16 3 14

66 5856 9 8596

76 4452

2 2 2

2

2

= +

= +

=

.

. ...

.

m

m

m

76 4452

8 743294574

8 74.

=

=

Label the hypotenuse for easy referencing

To calculate the length m, square root this value

Answer in square root form

Write full calculator reading before rounding

Approximate answer rounded to 2 decimal places

Rounded off decimal values are approximate answers only, so the '≈' symbol should be used.

3.14 units

`

Stop here if asked for answer in exact form

c a b2 2 2= +

units

units

`

`

`

`

` units

exact form

7ISERIES TOPIC



Where does it work? Your Turn Pythagoras’ Theorem


Complete these Pythagoras’ Theorem calculations to find the length of the hypotenuse in each triangle.

1

a

6

8

c

c

c

c

62 2

2

2

2= +

= +

=

c

c

=

=

8

15

g

b

g

g

g

82 2 2

2

2

= +

= +

=

g

g

=

=

exact form

Use Pythagoras’ Theorem to calculate the length of the hypotenuse in each of these triangles.2

a b

c

d

`

`

`

`

`

`

`

`

5

12

1.6

1.2

exact form

units units



7ISERIES TOPIC



3 Calculate the length of the hypotenuse in each of these triangles, leaving answers in exact form.

a b

4 Calculate the length of the hypotenuse in each of these triangles, rounding answers to 2 decimal places. psst! Remember to use the ‘≈’ for rounded answers.

a b

1.1

6.0

h12

35

n

10 units 9 units

c

3.4 units

5.9 units

p

7ISERIES TOPIC





Calculate the total length of the 3-stage flight path over the hills shown below accurate to 1 decimal place. psst! You need to do 3 hypotenuse calculations first.

5

CALCUL

ATING THE LENGTH OF THE HYPOTENUSE

..../...../2

0...c

ab

2

2

2

=+

198m

39m

36mStage 3

Flight path

Stage 1

Launch pad

252m40m 360m

Stage 2



7ISERIES TOPIC

Where does it work? Pythagoras’ Theorem

Calculating the length of a short side

To calculate a short side length in a right-angled triangle, the formula needs a little adjusting.

Subtract the given short side squared away from the hypotenuse squared.

Calculate the length of the missing side for this right-angled triangle.

Short side2 = hyponenuse2 - other short side2

Let’s label the short side ‘a’

15 unitsa

a

a

a

a

a

15 12

225 144

81

81

9

2 2 2

2

2

= -

= -

=

=

=

`

or a c b2 2 2= -

Always (longest side)2 – (smaller side)2

To calculate the length of a, square root this value


Calculate the length of side k for this right-angled triangle, leaving answer in exact form.


7.3

k

. .

. .

.

.

k

k

k

k

7 3 1 9

53 29 3 61

49 68

49 68

2 2 2

2

2

= -

= -

=

=

`

or k c a2 2 2= -

Label the hypotenuse for referencing

To calculate the length k, square root this value

Answer in exact form

Answers left in square root form are not approximations, so the ‘=’ can still be used.

12 units

1.9

a c b2 2 2= -

units

units

units`

`

`

7ISERIES TOPIC





Fill the gaps in these calculations to find the length of the missing short side in each triangle.1

a b

26

24

a

8.5

b1.3

5670

j

18.1 unitsa

18 unitsa b

a

a

a

262 2 2

2

2

= -

= -

=

.b

b

b

1 32 2 2

2

2

= -

= -

=

Use Pythagoras’ Theorem to calculate the length of the missing short side in each of these triangles.2

CALCUL

ATING THE LENGTH OF THE SHORT SI

DE ..../...../20...

ac

b2

2

2

=-

a

a

=

=

`

`

`

`

`

`

`

`

b

b

=

=units units



7ISERIES TOPIC



3 Calculate the length of the missing short side in each of these triangles, leaving answers in square root form.

11

17

b w

a b

a b

4 Calculate the length of the missing short side in each of these triangles, rounding answers to 1 decimal place.

psst! Remember to use the ‘. ’ for rounded off answers.

11.75

14.25

13.8

8.3

y

41.08

23.42

x

7ISERIES TOPIC




Combination of hypotenuse and short side calculations

Match the triangles with the correct side length on the right to reveal the missing answer.

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

=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

MI

E

H

Q

E

14.12

2.9

73.4

25

67.7

33

60

COMBO T

IME

CO

MBO TIME COMBO TIME ..../...../

20...

1 2 3 4 5 6

2

1

3

4

5

6



7ISERIES TOPIC

Where does it work? Pythagoras’ Theorem

Applications of Pythagoras’ Theorem

Distances that are difficult to measure can be solved using Pythagoras’ Theorem.

Calculate how far a 15 m support will reach up a wall if standing 9 m away from its base.

Let’s label the height up the wall ‘h’

15 m h

15 9

225 81

144

h

h

h

2 2 2

2

2

= -

= -

=

m12h =

Walls and buildings are perpendicular to the ground

This is a short side of a right-angled triangle

Square root this value to find h


Calculate the perimeter of the garden shaped like a right-angled triangle shown below.

35 12

1225 144

1369

c

c

c

2 2 2

2

2

= +

= +

=

m

m

c

c

1369

37

=

=

First need the distance along the hypotenuse

Square root this value

Add all the side distances together

Pythagoras’ Theorem is often used to calculate unknown lengths in perimeter and area calculations.

9 m

Remember: Perimeter is the total distance around the outside

`

`

`

`

`

m m m

m

35 12 37

84

= + +

=

` Perimeter of the garden

Support

35 m

c12 m

7ISERIES TOPIC





1 One end of a 13 m straight wire is attached to a flag pole 12 m above the ground. How far away from the base of the flag pole (x) will the other end be attached to the ground as a support?

3 To avoid going through a muddy swamp, Mila walks 1.7 km west and then 3.9 km South.

(i) How far is Mila away from where she started at the end of this walk? Round answer to 2 decimal places.

psst! West and South directions are perpendicular (90o) to each other.

APPLICATIONS

OF PY

THAGOR

AS’ THE

OREM

..../...

../20...


APPLICATIONS OF

(ii) How much further did Mila have to walk to avoid the swamp?

Start1.7 km

Finish

3.9 km

Gini has made a pudding in a large 42 cm by 34 cm tray. If she first cuts the pudding diagonally from one corner to the other, how long was the cut Gini made to the nearest whole cm?

2

42 cm

34 cm

x

12 m13 m



7ISERIES TOPIC


3.3 m


4 (i) Calculate the base length of the painted triangle below. (ii) Use the base length to calculate the area of the triangle. psst! The area equals (base # height) ' 2

5 The mouse wants to run the shortest path from point A to point C across the floor shown. Calculate the shortest path between these two points if corner B blocks the direct path.

2.6 m17 m

6 (i) Calculate the length of the side marked ‘y’. (ii) Calculate the perimeter of the trapezium.

(i)

(ii)

C

18 m

54.4 m

A

B

172 cm

120 cm

137 cm

y

(i)

(ii)

Diagram not drawn to scale.

Base length

7ISERIES TOPIC





Give these two trickier applications a go to earn an awesome passport stamp!

7 Use Pythagoras’ Theorem twice to find the distance between points X and Y. psst! Find the difference between WY and WX

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.

*

AWESO

ME *

..../.....

/20...

* AWESOME *

65

34

16

Y

X

W

A

C

B

6 m

18.5 m

9.5 m

D

Z



7ISERIES TOPIC

Pythagoras’ TheoremWhat else can you do?

Pythagorean triads

A Pythagorean triad is the special name given to a set of three positive integers that work in Pythagoras’ Theorem.

Pythagorean triad integers represent the side lengths of a right-angled triangle.

The integers 3, 4 and 5 form the best known Pythagorean triad.

5

Braces are used to display a set of integers

Integers work in Pythagoras’ Theorem

They form a right-angled triangle

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

?

?

24 8 22

576 64 484

576 548

2 2 2

!

= +

= +

You can use the

LHS = RHS test

approach here too

Pythagoras’ Theorem is used to show if a set of three integers form a Pythagorean triad.

Test: does

4

Remember: integers are just whole numbers.

` Is , ,8 22 24" , a Pythagorean triad?

, ,3 4 5" , is a Pythagorean triad because 3 4 52 2 2+ =

, ,3 4 5" ,

3

Because each integer is a side length for a right-angled triangle, negative values are not allowed.

Test to see if: (largest value)2 = (smallest value)2 + (middle value)2

Largest value

Middle value

Smallest value

(i) , ,8 22 24" , (ii) , ,9 12 15" ,

Largest value

Middle value

Smallest value

Test: does ?

?

15 9 12

225 81 144

225 225

2 2 2= +

= +

=

` Is , ,9 12 15" , a Pythagorean triad?

Yes No Yes No

15

12

9

=

=

=

24

22

8

=

=

=

7ISERIES TOPIC



What else can you do? Your Turn Pythagoras’ Theorem

Pythagorean triads

1 Write the side lengths of these right-angled triangles as a Pythagorean triad set.

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

PYTHAGOREA

N TRIADS PYTHAGOREAN TRIADS ..../...

../20...3

5

4

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" , { , , }

{ , , }

{ , , }

a



7ISERIES TOPIC

What else can you do? Pythagoras’ Theorem

Euclid’s formula for Pythagorean triads

Euclid of Alexandria (a Greek mathematician) developed this method to find most Pythagorean triads:

Step 1: Choose two positive integers p and q. When you pick these integers, make p larger than q i.e. p > q

Step 2: Substitute values for p and q into these to make a Pythagorean triad:

small integer other small integer largest integer

, 2 ,p q pq p q2 2 2 2- +" ,

Use the values p = 3 and q = 2 to make a Pythagorean triad.

Substitute in p and q values

Calculate final values

Integers form a Pythagorean triad

Use Euclid’s formula to make a Pythagorean triad that contains the number 8.

Here is another example with a specific request.

For the values p = 3 and q = 2

Let’s check that it works

`

, 2 ,p q pq p q2 2 2 2- +" ,

, ,9 4 12 9 4- +" ,

, ,5 12 13" ,

5 12 13 ?2 2 2+ =

25 144 169?+ =

169 169=

3 2 , 2 3 2 , 3 22 2 2 2# #- +" ,

This will be the easiest to use this time

p > q and ensures a value of 8

Substitute in p and q values

Calculate final values

Put values into ascending order for triad

, 2 ,p q pq p q2 2 2 2- +" ,

4 1 , 2 4 1, 4 12 2 2 2# #- +" ,

, ,15 8 17" ,

, ,8 15 17" ,

pq2 8=Let

` p = 4 and q = 1

, ,16 1 8 16 1- +" ,

7ISERIES TOPIC





1 Complete this table for the given values of p and q to make Pythagorean triads.

2 Make Pythagorean triads matching each of these specific requests.

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

EU

CLID’S FORMULA FOR PYTHAGOREAN

TRAIDS

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

,,

pq

pq pq

2

22

22

-

+

"

,

*

(ii) Find a Pythagorean triad in which q = 6 and p q2 2+ is equal to 61.

p2 - q2 p 2pq p2 + q2 Triadq

2 1

3 1

5 2

7 6

11 3

21 18



7ISERIES TOPIC


Remember me?


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

Use the space below to show why the value of p must be greater than the value of q when using Euclid’s formula to find a Pythagorean triad.

Use your own values of p and q to help show your answer.

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

This is definitely worth an awesome stamp!!

3

4

* AWESOME *

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

* AWESOME

*

7ISERIES TOPIC



What else can you do? Pythagoras’ Theorem

Wheel of Theodorus

When squares are drawn using each side length of a right-angled triangle, something interesting happens.

How does this work?

Using Pythagoras’ Theorem:

Starting with this isosceles triangle, each new right-angled triangle is built using the hypotenuse of the previous one.

The length of the longest sides form a nice square root number pattern.

c a b

c a b

2 2 2

2 2

= +

= +`

` for the first triangle:

` for the second triangle:

c 1 1

2

2 2= +

=

( )c 1 2

1 2

3

2 2= +

= +

=

c

2

1 1

3

2

12

1

1 c

c

b

a

2

3

4 5

6

7

8

9

10

11

12

1

1

1 1

1

1

1

1

1

1

1

The pattern continues in this fashion always using 1 as the shortest side value.

1

1



7ISERIES TOPIC


Wheel of Theodorus

Using the start made for you, continue the pattern always using 2 as the shortest side value to create your own neat spiral wheel.

8

WHEEL OF

THEODORUS WHEEL OF THEODORUS ..../

...../20..

.

2

2

2

7ISERIES TOPIC




Reflection Time

Reflecting on the work covered within this booklet:

1 What useful skills have you gained by learning Pythagoras’ Theorem?

2 Write about one way you think you could apply Pythagoras’ Theorem to a real life situation.

3 If you discovered or learnt about any shortcuts to help with Pythagoras calculations or some other cool facts, jot them down here:



7ISERIES TOPIC

Cheat Sheet Pythagoras’ Theorem

Here is a summary of the things you need to remember for Pythagoras’ Theorem


• These special triangles all have a right-angle (angle of size 90o) as one of the internal angles.

• The 90o angle is always opposite the longest side.

• The two shorter sides are always perpendicular to each other.


• The squares and right-angled triangles section showed that a relationship exists between the side lengths of right-angled triangles. This relationship is called Pythagoras’ Theorem.

• If the rule does not work, then it is not a right-angled triangle.


• Use Pythagoras’ Theorem to calculate the length of the hypotenuse if the two shorter sides of the right-angled triangle are already known.

• The order that we put the short side values into the formula does not matter.


• To calculate a short side length in a right-angled triangle, the formula needs a little adjusting.

• Subtract the given short side squared away from the hypotenuse squared.

Pythagorean Triads

• A Pythagorean triad is a set of three positive numbers that work in Pythagoras’ Theorem.

Making Pythagorean Triads

Step 1: Choose two positive numbers p and q.

When you pick these numbers, make p larger than q i.e. p > q

Step 2: Substitute values for p and q into this to make a Pythagorean triad: , 2 ,p q pq p q2 2 2 2- +" ,

Step 3: Write the values in ascending order.

cother short sidehypotenuse

short side a

b

(short side)2 + (other short side)2 = (longest side)2 a2 + b2 = c2

always the hypotenuse

a c b2 2 2= -

c a b2 2 2= +

7ISERIES TOPIC



Squares and right-angled triangles: Jigsaw Puzzle

Step 1: Cut the two shaded squares out from the page.

Step 2: Cut the larger of these two along the dotted lines.

Step 3: Arrange all the pieces to fit perfectly inside this square.

Step 4: Stick the pieces to the page to show the area of the two smaller squares add together to give the area of this square on the hypotenuse.

Jigsaw Puzzle Pythagoras’ Theorem



7ISERIES TOPIC

APPLICATIONS

OF

PYTHA

GORAS’

THEORE

M

..../...

../20...


APPLICATIONS OF

RIGHT-ANGLED

TRIANGLE

S 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-

+

",

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Pythagoras Mathaletics

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Transcript of Pythagoras Mathaletics