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http://www.mathsisfun.com/fractions.html - PPT Presentation

Year 9 Mathematics Pythagoras Theorem Learning Intentions To correctly label a rightangled triangle to understand the relationship between the sides of right angledtriangle to solve problems using Pythagoras ID: 728365

hypotenuse triangle side angled triangle hypotenuse angled side theorem pythagoras pythagoras

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Slide1

http://www.mathsisfun.com/fractions.html

Year 9 Mathematics

Pythagoras TheoremSlide2

Learning Intentions

To correctly label a right-angled triangleto understand the relationship between the sides of right angled-triangleto solve problems using Pythagoras’ TheoremSlide3

Areas

Your teacher will give you a copy of this diagram.Cut out the diagram along the lines carefully.Not try rearranging the pieces from squares B and C on top of square A.

What do you notice?Slide4

The Result!

When Pythagoras did this he noticed that:The area of the square made along side A was the same as the sum of the areas on sides B and C.Pythagoras stated this as:The square on hypotenuse is equal

to the sum of the squares on the other two sides.Be careful – this only works for right-angled triangles!Slide5

Dissections!

Henry Perigal (1801 – 1898) created a visual demonstration of Pythagoras’ Theorem.Slide6

Where is the Hypotenuse

Pythagoras called the longest side in the triangle the Hypotenuse.

HypotenuseSlide7

The Algebra

Pythagoras created an equation to help him solve problems involving right angled triangles.He wrote:a2 = b

2 + c2But make sure you label the sides correctly!

a

b

cSlide8

Writing the Formula

Write out Pythagoras’ Theorem for the following triangles:Slide9

How Do I Use It?

Calculate the size of the hypotenuse in this diagram. x2 = 15

2 + 82 = 225 + 64 = 289 x = 289

x = 17 cm15

cm

8 cm

xSlide10

Do that again

Calculate the size of the hypotenuse in this diagram. x2 = 3.6

2 + 4.82 = 12.96 + 23.04 = 36 x =

36 x = 6 cm

3

.6 cm

x

cm

4.8Slide11

Not the Hypotenuse!

You need to be careful if you are finding the length of the side that is not the hypotenuse. 152

= 92 + x2 225 = 81 + x2

144 = x2 144 = x x = 12 cm

9

cm

15

cm

xSlide12

Pythagorean Triples

There are some special right angles triangles.3, 4, 5In this triangle the sides are all whole numbers.We can also enlarge this triangle to:

6, 8, 10 or 9, 12, 15Or reduce it to:1.5, 2, 2.5

25

9

16Slide13

Ancient Egypt

The Egyptians new about the 3, 4, 5 triangle. They

were able to use this knowledge in the construction of pyramids, temples and other buildings to ensure a perfect right-angle at the corners.They were in fact using the what was to become Pythagoras’ Theorem, 1500 years later.Rope with 12 equally

spaced knots.Slide14

Pythagorean Triples

5, 12, 13We can also enlarge this triangle to:10, 24, 26 or 15, 26, 39

Or reduce it to:2.5, 6, 6.5

169

144

25Slide15

Pythagorean Triples

7, 24, 25We can also enlarge this triangle to:14, 48, 50 or 21, 72, 75

Or reduce it to:3.5, 12, 12.5Recognising these triples can help to find the answers without using a calculator!

625

576

49Slide16

QuestionsSlide17

Is it a Right Angled Triangle?

Sometimes you are asked if a triangle is right angled.To do this you must use Pythagoras’ theorem and check if it is correct.

187

11

17

815Slide18

Applications

A skier travels down a steep slope loosing a vertical height of 560m as he covers 720 horizontally. Calculate the actual distance he skis down the slope.Slide19

Applications

A 12m ladder rests against the side of a house. The top of the ladder is 9.5m from the floor. How far is the base of the ladder from the house?

12 m

9.5 m

LSlide20
Slide21

Applications

The diagram shows an Olympic BMX. The frame of the bike is made from lightweight aluminium. The frame is shown in green.Calculate the total length of tubing needed to make the frame. Give your answer ain centimetres to 3 significant figures.

Aluminium tubing weight 800g per metre. Work out the total mass of the frame in kg, to 3 significant figures.Slide22

Additional Questions

How long is the diagonal of a square with a side of 6m?A ship leaves port and sails directly to a lighthouse 20 km east and 15 km north. How far has it sailed?Calculate the height of an equilateral triangle with a side length of 6 cm.