MTH 4-16a 1-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem Squaring a Number and Square Roots Investigating Pythagoras.

MTH 4-16a

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

Pythagoras Pythagoras TheoremTheorem

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Investigating Pythagoras Theorem

Finding the length of the smaller side

Calculating the Hypotenuse

Solving real-life problems

Distance between two points & Mixed Problems

MTH 4-16a


Starter Questionsw

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2 2

.

3 4

I s 14 2 the same as Explain?40 5

1. I s (- 2) 3 3 = 3. Explain your answer

2. Calculate 45% of 360

3.

4. Calculate

MTH 4-16a


Squaring a Squaring a NumberNumber

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1.1. To understand what is To understand what is meant by the termmeant by the term

‘‘squaring a number’squaring a number’

1. We are learning the term ‘‘squaring a number’.squaring a number’.

2.2. Be able to calculate Be able to calculate squares both mentally and squares both mentally and using the calculator.using the calculator.

MTH 4-16a

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

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To square a number means to :

“Multiply it by itself”

means 9 x 9 = 81

Example :

29

means 10 x 10 = 100210

MTH 4-16a


Now try Exercise 1Ch59 (page 202)

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Square Root of a Square Root of a numbernumber

You now know how to find :

We can ‘undo’ this by asking

““which number, times itself, gives 81”which number, times itself, gives 81”

92 = 9 x 9 = 81

From the top line, the answer is 9

This is expressed as : “the SQUARE ROOT of 81 is 9”“the SQUARE ROOT of 81 is 9”

or in symbols we write : 81=9

MTH 4-16a



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Square Root of a Square Root of a NumberNumber

19 Apr 2023Compiled by Mr. Lafferty Maths

Dept.

2 2 = 41 where a = 4 and b = (-5). Explain?I s a b

Starter QuestionsStarter Questions

Q1.Q1. Are the missing anglesAre the missing angles6565oo, 40, 40oo and 65 and 65oo

Q2.Q2. CalculateCalculate

Q3. Q3. The cost of a new computer is £1000 +vat. If the vat is The cost of a new computer is £1000 +vat. If the vat is charged at 12% what is the total cost.charged at 12% what is the total cost.

Q4. Q4. The cost of a bag of sugar is £1.12. How much 50 bags The cost of a bag of sugar is £1.12. How much 50 bags cost.cost.

115o

NON-CALCULATORNON-CALCULATOR

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MTH 4-16a


Dept.

‘‘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

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MTH 4-16a


Dept.


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

34

5

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MTH 4-16a


Dept.

Right – Angle TrianglesRight – Angle Triangles

c

b

a



68

10

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Dept.


c b

a



5

12

13

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MTH 4-16a


Dept.


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

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MTH 4-16a


Dept.

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

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MTH 4-16a


Dept.

Pythagoras’s Pythagoras’s TheoremTheorem

a

bc

2 2 2a b c

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MTH 4-16a


Dept.

Summary of Summary of Pythagoras’s TheoremPythagoras’s Theorem

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

2 2 2a b c

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MTH 4-16a

MTH 4-16a


Now try Exercise 359(page 204)

Pythagoras Theorem

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MTH 4-16a


Starter Questionsw

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1. I s 4 - 3 4 = 4. Explain

2. Write 88% as a decimal

3. Why is the LCM of 12 and 7 equal to 84.

MTH 4-16a


Calculating Calculating HypotenuseHypotenuse

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1. Remember the term hypotenuse “ the longest side”

1. We are learning to use Pythagoras Theorem to calculate the length of the hypotenuse

“the longest side”2. Apply Pythagoras Theorem

to calculate the hypotenuse.


Dept.

Calculating the HypotenuseCalculating the Hypotenuse

8

12

c

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

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

2c =208

c = 208 =14.42km

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Example 1MTH 4-16a


Dept.

Calculating the Hypotenuse

Aeroplaneb = 8

a = 15

c

Lennoxtown

Airport

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?

2 2 2c =a +b2 2 2c =15 +8

2c =289

c = 289 =17km

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Example 2MTH 4-16a

MTH 4-16a



Calculating Calculating HypotenuseHypotenuse

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MTH 4-16a


Starter Starter QuestionsQuestions

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24 , 3 2x y and z z xy

1. Calculate 0.5% of £45

2. I f . Find

3. Calculate 60% of £360

4. Solve 2x+3 = x + 11

MTH 4-16a


Solving Real-Life Solving Real-Life ProblemsProblems

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1.1. Be able to solve real-life Be able to solve real-life problems using problems using Pythagoras Theorem Pythagoras Theorem showing clear working.showing clear working.

1. We are learning how Pythagoras Theorem can be used to solve real-life problems.

MTH 4-16a



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

What is the length of the rod?

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

2 2 2c =a +b2 2 2c =8 +152c =289

c = 289 =17m

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15m

8m

rod

MTH 4-16a



Example 2A garden is rectangular in shape. A fence is

to be put along the diagonal as shown below. What is the length of the fence.

2 2 2c =a +b2 2 2c =10 +152c =325

c = 325 =18.03m

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10m

15m

MTH 4-16a




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MTH 4-16a


Starter Questionsw

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5 , 2 4

27

(2 4 )

x

x y and z y z x

1. Solve

2. I f . Find

3. Calculate 80% of £700

4. Find the median, mode and mean for

6, 7, 6, 4, 3, 10

MTH 4-16a


Length of the smaller Length of the smaller sideside

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1.1. Apply Pythagoras Apply Pythagoras Theorem to find the Theorem to find the length of smaller side length of smaller side showing clear working.showing clear working.

1. We are learning how Pythagoras Theorem can be used to find the length of the smaller side.

MTH 4-16a



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

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20cm 12cm

a cm

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

than hypotenus

e

MTH 4-16a



Example : Find the length of side b ?

2 2 2c =a +b

2 2 2b =10 - 82b =36

b = 36 =6cm

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10cm b cm

8 cm

2 2 2b =c - a

Check answer ! Always smaller

than hypotenus

e

MTH 4-16a



Length of smaller Length of smaller sideside

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MTH 4-16a

Starter Questionsw

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ALWAYS comes up in

exam !!

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1.1. Apply Pythagoras Theorem Apply Pythagoras Theorem to find length of a line.to find length of a line.

1. We are learning how Pythagoras Theorem can be used to find the length of a line.

2.2. Show all working.Show all working.

Finding the Finding the Length of a LineLength of a Line

MTH 4-16a

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

(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

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

MTH 4-16a



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Finding hypotenuse c2 2 2c =a +b

2 2 2a =c - b

2 2 2b =c - a

Finding shorter side a

Findingshorter side b

a

c b

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

MTH 4-16a




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MTH 4-16a 1-Aug-15Compiled by Mr. Lafferty Maths Dept. Pythagoras Theorem Squaring a Number and Square Roots Investigating Pythagoras.

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