1-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem Investigating Pythagoras Theorem Finding the length of the smaller.

19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.

Pythagoras Pythagoras TheoremTheorem

ww

w.m

ath

srevis

ion

.com Investigating Pythagoras Theorem

Finding the length of the smaller side

Solving Real – Life Problems

Pythagoras Theorem Twice

Converse of Pythagoras Theorem

S3 Credit

19 Apr 2023 Compiled by Mr. Lafferty Maths Dept.

Starter QuestionsStarter Questions

Q1.Q1. Explain why 4 + 6 Explain why 4 + 6 xx 5 = 34 and not 50 5 = 34 and not 50

Q2.Q2. CalculateCalculate

2 2 where a = 4 and b = (-5)a b

Q3. Q3. Does xDoes x22 – 121 factorise to (x – 11) (x - 11) – 121 factorise to (x – 11) (x - 11)

Q4. Q4. The cost of an iPod is £80 including VAT. The cost of an iPod is £80 including VAT. How much is the iPod BEFORE VAT.How much is the iPod BEFORE VAT.

NON-CALCULATORNON-CALCULATOR

ww

w.m

ath

srevis

ion

.com


‘‘To investigate the right-angle triangle and to come To investigate the right-angle triangle and to come up with a relationship between the lengths of its two up with a relationship between the lengths of its two shorter sides and the longest side which is called the shorter sides and the longest side which is called the hypotenuse. ‘hypotenuse. ‘

c

b

a

Aim of today's LessonAim of today's Lesson

Right – Angle Right – Angle TrianglesTriangles

ww

w.m

ath

srevis

ion

.com



c

b

a

What is the length of a ?

What is the length of b ?

Copy the triangle into your jotter and measure the length of c

3

4

5

ww

w.m

ath

srevis

ion

.com


Right – Angle TrianglesRight – Angle Triangles

c

b

a




6

8

10

ww

w.m

ath

srevis

ion

.com



c b

a




5

12

13

ww

w.m

ath

srevis

ion

.com



a b c a2 b2 c2

3 4 5

5 12 13

6 8 10

c

b

a

Copy the table below and fill in the values that are missing

ww

w.m

ath

srevis

ion

.com


Right – Angle TrianglesRight – Angle Triangles

2 2 2a b c

a b c a2 b2 c2

3 4 5 9 16 25

5 12 13 25 144 169

6 8 10 36 64 100

Can anyone spot a

relationship between a2, b2, c2.

c

b

a

ww

w.m

ath

srevis

ion

.com


Pythagoras’s Pythagoras’s TheoremTheorem

a

bc

2 2 2a b c

ww

w.m

ath

srevis

ion

.com


Pythagoras’s Pythagoras’s TheoremTheorem

y

x

z

2 2 2( ) ( ) ( )xy yz xz

ww

w.m

ath

srevis

ion

.com


Summary of Summary of Pythagoras’s TheoremPythagoras’s Theorem

Note: The equation is ONLY valid for right-angled triangles.

2 2 2a b c

ww

w.m

ath

srevis

ion

.com


Calculating Calculating HypotenuseHypotenuse

ww

w.m

ath

srevis

ion

.com Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Know the term Know the term hypotenuse hypotenuse “ the “ the longest side”longest side”

1. Use Pythagoras Theorem to calculate the length of the hypotenuse

““the longest side”the longest side” 2.2. Use Pythagoras Theorem Use Pythagoras Theorem to calculate the to calculate the hypotenuse.hypotenuse.

S3 Credit



ww

w.m

ath

srevis

ion

.com

Two key points when dealing with right-angled triangles

The longest side in a right-angled triangle is calledThe HYPOTENUSE

The HYPOTENUSE is ALWAYS opposite the right angle

a

b cc2 = a2 + b2

y

x

z

(xz)2 = (xy)2 + (yz)2

S3 Credit


Calculating the HypotenuseCalculating the Hypotenuse

8

12

c

Q2.Q2. Calculate the longest length of the right-Calculate the longest length of the right-angled triangle below.angled triangle below.

2 2 2c =a +b2 2 2c =12 +8

2c =208

c = 208 =14.42km

ww

w.m

ath

srevis

ion

.com

Example 1Example 1


Calculating the HypotenuseCalculating the Hypotenuse

LA = 8

GL = 15

Lennoxtown

Airport

Q1.Q1. An aeroplane is preparing to land at Glasgow An aeroplane is preparing to land at Glasgow Airport. Airport. It is over Lennoxtown at present which is It is over Lennoxtown at present which is 15km from 15km from the airport. It is at a height of 8km. the airport. It is at a height of 8km.

How far away is the plane from the airport?How far away is the plane from the airport?

( )GA 2 2 2=(GL) +(LA)

( )GA 2 2 2=15 +8

( )GA 2 =289

GA = 289 =17km

ww

w.m

ath

srevis

ion

.com

Example 2Example 2

A

LG


Now try Ex 2.1 and 2.2MIA Page 147


ww

w.m

ath

srevis

ion

.com

S3 Credit


S3 Starter QuestionsS3 Starter Questionsw

ww

.math

srevis

ion

.com

2 2

5

(2 3 4)

8 4 (2

4 45 115 9

4 1)

3 65 25

1. Does

2. Expand

3. Does factorise to

4.

y y x

d d dd

S3 Credit


Length of the smaller Length of the smaller sideside

ww

w.m

ath

srevis

ion


1.1. Use Pythagoras Use Pythagoras Theorem to find the Theorem to find the length of smaller side.length of smaller side.

1. To show how Pythagoras Theorem can be used to find the length of the smaller side.

2.2. Show all working.Show all working.

S3 Credit


Example : Find the length of side a ?


To find the length of the smaller side of a right-angled triangle

we simply rearrange Pythagoras Theorem.

2 2 2c =a +b

2 2 2a =20 -122a =256

a= 256 =16cm

ww

w.m

ath

srevis

ion

.com

20cm 12cm

a cm

2 2 2a =c - bCheck answer ! Always smaller

than hypotenus

e

S3 Credit


Example : Find the length of side a ?


2 2 2c =a +b

2 2 2b =10 - 82b =36

b = 36 =6cm

ww

w.m

ath

srevis

ion

.com

10cm b cm

8 cm

2 2 2b =c - a

Check answer ! Always smaller

than hypotenus

e

S3 Credit

A

B

C D E F

G H

IJ

K L

P(x,y)

o

r

M

N

O

P

Q R

S

T

U V

W Z

AC


Now work throughEx3.1 and Ex 3.2

Odd Numbers Only

Length of smaller Length of smaller sideside

ww

w.m

ath

srevis

ion

.com

S3 Credit



ww

.math

srevis

ion

.com

-4

2

45 2.56 10

65% £40 £26

3 1 16 14 4 16

- 6 8

1. Calculate

2. Write in scientific notation

3. Factorise

4. Explain why

k

of

k

S3 Credit


Solving Real – Life Solving Real – Life ProblemsProblems

Using Pythagoras Using Pythagoras TheoremTheorem

ww

w.m

ath

srevis

ion


1.1. Apply Pythagoras Theorem Apply Pythagoras Theorem to solve real-life problems.to solve real-life problems.

1. To show how Pythagoras Theorem can be used to solve real-life problems.

2.2. Show all working.Show all working.

S3 Credit


Example : A steel rod is used to support a treewhich is in danger of falling down.

What is the height of the tree ?

Solving Real-Life Solving Real-Life ProblemsProblems

When coming across a problem involving When coming across a problem involving finding a missing side in a right-angled triangle, finding a missing side in a right-angled triangle, you should consider using Pythagoras’ Theorem you should consider using Pythagoras’ Theorem to calculate its length.to calculate its length.

2 2 2c a b

2 2 217 - 8a

2 225a

225 15a m

ww

w.m

ath

srevis

ion

.com

17m

8m

rod

b

ac

S3 Credit



Example 2Example 2A garden has a fence around its perimeter and A garden has a fence around its perimeter and along its diagonal as shown below. along its diagonal as shown below. What is the length of the fence from D to C.What is the length of the fence from D to C.

2 2 2( ) ( ) ( )AC DC AD

2 2 2( ) 13 - 5DC

2( ) 144DC

144 12DC m

ww

w.m

ath

srevis

ion

.com

5m

b m

13m

A B

CD

S3 Credit


Now work throughEx4.1 and Ex 4.2

Odd Numbers Only

Length of smaller Length of smaller sideside

ww

w.m

ath

srevis

ion

.com

S3 Credit



ww

.math

srevis

ion

.com

2 2

2

4 , 3 2

24 16

3 1 15 2 74 8 8

2

1. True or false

2. I f .

Find

3. Factorise

p q and r

q

y y

z r

S3 Credit



TwiceTwice

ww

w.m

ath

srevis

ion


1.1. Use the appropriate Use the appropriate form of Pythagoras form of Pythagoras Theorem to solving Theorem to solving harder problems.harder problems.

1. To use knowledge already gained on Pythagoras Theorem to solve harder problems using Theorem twice. 2.2. Show all working.Show all working.

S3 Credit

19 Apr 202319 Apr 2023


Problem : Find the length of h.

2 2 2( ) ( ) ( )BD AD AB

2 2 2( ) 15 13BD

2( ) 394BD

394BD

ww

w.m

ath

srevis

ion

.com

h

A

B C

D

12

13

15

Find length BD first

S3 Credit

19.85BD

19.85

19 Apr 202319 Apr 2023


Problem : Find the length of length y.

2 2 2( ) ( ) ( )BC BD DC

2 2 2( ) ( )h BD DC

2 2 2(19.85) (12)h

2 394 144h

ww

w.m

ath

srevis

ion

.com

h

A

B C

D

12

13

15

Now find h

538h

23.2 ( 1 . .)h to d p

S3 Credit

394

19 Apr 202319 Apr 2023


Problem : Find the diagonal length of the cuboid AG.

2 2 2( ) ( ) ( )AH AD DH

2 2 2( ) 8 6AH

2( ) 64 36AH

100AH

ww

w.m

ath

srevis

ion

.com

Find AH first

10AH cm A

B C

D

E

F G

H6cm

8cm

7cm

10cm

S3 Credit

19 Apr 202319 Apr 2023


Problem : Find the diagonal length of the cuboid AG.

2 2 2( ) ( ) ( )AG AH HG

2 2 2( ) 10 7AG

2( ) 100 49AG

2( ) 149AG

ww

w.m

ath

srevis

ion

.com

Now find AG

12.2 ( 1 . .)AG cm to d p

S3 Credit

A

B C

D

E

F G

H6cm

8cm

7cm

10cm


Now try Ex 5.1Ch8 (page 154)


ww

w.m

ath

srevis

ion

.com

S3 Credit



ww

.math

srevis

ion

.com

2 2 2 2

(1 )(2 3 )

1

1 1 16 7 42

- ( )

2 515 3

1. True or false

2. Expand

3. Does factorise to

4.

w w

ab ba b a a b

S3 Credit


Converse of Converse of Pythagoras Pythagoras

TheoremTheorem

ww

w.m

ath

srevis

ion


1.1. Apply the converse of Apply the converse of Pythagoras Theorem to Pythagoras Theorem to prove a triangle is prove a triangle is right-angled.right-angled.

1. To explain the converse of Pythagoras Theorem to prove a triangle is right-angled.

S3 Credit

2. Right-angle is directly opposite C.



TheoremTheorem

ww

w.m

ath

srevis

ion

.com

2 2 2a +b =cb

a

Converse Theorem states that if

S3 Credit

1. Then triangle MUST be right-angled.

c

Hypotenuse

Converse1 – talk

Converse2 – opposite, reverse



TheoremTheorem

ww

w.m

ath

srevis

ion

.com

Problem : Is this triangle right-angled ? Explain Answer

2 2 2a +b =c

2c =100

117 100

10cm

6 cm

9 cm

2 2 2 2a +b =9 +6 =81+36 = 117

If it is then Pythagoras Theorem will be true

By the Converse Theorem, triangle is NOT right-angled

S3 Credit



TheoremTheorem

ww

w.m

ath

srevis

ion

.com

Problem : A picture frame manufacturer claims that his are rectangular is his claim true.

2 2 2a +b =c

2c =25002 2 2a +b =c 30 cm

40 cm2 2 2 2a +b =40 +30 =1600 +900 = 2500

If it is then Pythagoras Theorem will be true

By the Converse Theorem, frame IS rectangular

50cm

S3 Credit


Now try Ex 6.1 & 6.2

Ch8 (page 156)


TheoremTheorem

ww

w.m

ath

srevis

ion

.com

S3 Credit

1-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem Investigating Pythagoras Theorem Finding the length of the smaller.

Documents

lafferty maths

length of c

length of b

longest length

calculating hypotenuse

use pythagoras theorem

rightangled triangles

term hypotenuse