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15 Jun 2022 15 Jun 2022 Compiled by Mr. Lafferty Maths Compiled by Mr. Lafferty Maths Dept. Dept. Pythagoras Pythagoras Theorem Theorem www.mathsrevision.com Revision of simple Pythagoras Theore Converse of Pythagoras Theorem 3D - Pythagoras Theorem Exam Questions Nat 5
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15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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Page 1: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Pythagoras Pythagoras TheoremTheorem

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Revision of simple Pythagoras Theorem

Converse of Pythagoras Theorem

3D - Pythagoras Theorem

Exam Questions

Nat 5

Page 2: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Starter QuestionsStarter Questionsw

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Nat 5

Given the two straight lines

3y + 2x - 12 = 0 4x + 6y = -24

In pairs explain what they have in commonAnd where each cross the x-axis and y-axis.

Page 3: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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

1. We are revising Pythagoras Theorem

2.2. Use Pythagoras Theorem Use Pythagoras Theorem to calculate both the to calculate both the hypotenuse and smaller hypotenuse and smaller sides of a right-angled sides of a right-angled triangle.triangle.

Nat 5

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

Page 4: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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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

Nat 5

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

Page 5: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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

What is the height of the tree ?

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

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17m

8m

rod

b

ac

Nat 5

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

Page 6: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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

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5m

b m

13m

A B

CD

Nat 5

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

Page 7: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

MTH 4-16a

19 Apr 2023Created by Mr. Lafferty Maths

Dept.w

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1 32 540

1

3

2

5

4

1 32 40

1

32

54

6

5 76 8 9 10

78

(7,7)

(2,4)

3

5

Finding the Length of a Line

Discuss with your partner how we might find the length of the

line.

2 2 2c a b

2 2 25 3c

25 9c

34c

5.83c

Page 8: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

19 Apr 2023Created by Mr. Lafferty Maths

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1 32 540

1

3

2

5

4

1 32 40

1

32

54

6

5 76 8 9 10

78

(9,1)

(0,6)

5

9

2 2 2c a b

2 2 25 9c

25 81c

106c

10.3c

Pythagoras Theorem to find the length of a LineMTH 4-16a

Page 9: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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h

A

B C

D

12

13

15

Find length BD first

Nat 5

19.85BD

19.85

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

Page 10: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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h

A

B C

D

12

13

15

Now find h

538h

23.2 ( 1 . .)h to d p

Nat 5

394

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

Page 11: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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Nat 5

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

NOTE !!!!

Pythagoras Theorem tends to pop up within circle type question

Page 12: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 13: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Page 14: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Now try N5 TJEx5.1

Ch5 (page 44)

Revision of Pythagoras Revision of Pythagoras TheoremTheorem

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Nat 5

Page 15: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Starter QuestionsStarter Questionsw

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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

Nat 5

Page 16: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Converse of Converse of Pythagoras Pythagoras

TheoremTheorem

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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. We are learning the converse of Pythagoras Theorem to prove a triangle is right-angled.

Nat 5

Page 17: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

2. Right-angle is directly opposite C.

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

Converse of Converse of Pythagoras Pythagoras

TheoremTheorem

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2 2 2a +b =cb

a

Converse Theorem states that if

Nat 5

1. Then triangle MUST be right-angled.

c

Hypotenuse

Converse1 – talk

Converse2 – opposite, reverse

Page 18: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Converse of Converse of Pythagoras Pythagoras

TheoremTheorem

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Problem : Is this triangle right-angled ? Explain Answer

2c =10210cm

6 cm

9 cm

2 2a +b

If it is then Pythagoras Theorem will be true.

Nat 5

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

2c =100

Since 117 ≠ 100 then by the Converse Theorem of

Pythagoras triangle IS NOT right-angled.

a

b

c

Page 19: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Converse of Converse of Pythagoras Pythagoras

TheoremTheorem

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

30 cm

40 cm

If it is then Pythagoras Theorem will be true

50cm

Nat 5

2c =5022 2a +b2 2= 30 +40

=900 +1600 = 2500

2c =2500

Since 2500 = 2500 then by the Converse Theorem of Pythagoras triangle IS

right-angled.

a

b

c

Page 20: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Now try N5 TJEx5.2

Ch5 (page 46)

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Nat 5

Converse of Converse of Pythagoras Pythagoras

TheoremTheorem

Page 21: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Starter QuestionsStarter Questionsw

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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

Nat 5

Page 22: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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1.1. Use the appropriate Use the appropriate form of Pythagoras form of Pythagoras Theorem to solving 3D Theorem to solving 3D problems.problems.

1. We are learning how to apply knowledge already gained on Pythagoras Theorem to solve 3D problems.

2.2. Show all working.Show all working.

Nat 5

3D Pythagoras 3D Pythagoras TheoremTheorem

Page 23: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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Find AH first

10AH cm A

B C

D

E

F G

H6cm

8cm

7cm

10cm

Nat 5

3D Pythagoras 3D Pythagoras TheoremTheorem

Page 24: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

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Now find AG

12.2 ( 1 . .)AG cm to d p

Nat 5

A

B C

D

E

F G

H6cm

8cm

7cm

10cm

3D Pythagoras 3D Pythagoras TheoremTheorem

Page 25: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.

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

Now try N5 TJEx5.3

Ch5 (page 47)

3D Pythagoras 3D Pythagoras TheoremTheorem

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Nat 5

Page 26: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 27: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 28: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 29: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 30: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 31: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 32: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 33: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 34: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
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Page 42: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 43: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 44: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.
Page 45: 15-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem  Revision of simple Pythagoras Theorem Converse of Pythagoras Theorem.