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
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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
LSlide20Slide21
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.