Contents(c) Introduction of Pythagoras’ Theorem...................................................................................2

(d) Two ways to prove Pythagoras' Theorem............................................................................3

(e) The similarities and differences used in (d)..........................................................................4

(f) Usages of Pythagoras’ Theorem in daily life.......................................................................5

(g) Conclusion............................................................................................................................6

(h) Self-reflection.......................................................................................................................6

(i) Reference...............................................................................................................................7

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(c) Introduction of Pythagoras’ Theorem

Over 2000 years ago there was a discovery about right angled triangles. It is called the Pythagorean Theorem, also known as Pythagoras's theorem. It is a relation among the three sides of a right triangle. If squares are made on each of the three sides, the biggest square is equal to the other two square put together. This is called the ‘Pythagoras’ Theorem’. It was named after the Greek mathematician Pythagoras. It is one of the most famous Pythagorean proposition.

…..Note that c is the longest side (hypotenuse) while a and b are the shorter sides…..

The “Pythagorean equation “can be calculated in the formula a2 + b2= c2.

Therefore, the sides of the triangle can be deducted from the formula a2 + b2 =c2:

If the length of the hypotenuse c and of one side (a or b) are known, then the length of the other side can be calculated as

Or

According to this image, we can know that the triangle has a right angle as 42 + 32

= 16 + 9=25

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(d) Two ways to prove Pythagoras' Theorem

Proof #1

First, we start with 4 triangles of the same size. Three of it has been rotated 90°, 180°, and 270°, respectively. Each of it has area ab/2.

Secondly, put them together to form a big square with side c and a small square in the middle.

Each side of the small square is (a-b).So the area of the small square is (a - b)² and the area of the 4 triangles is 4ab/2, which is 2ab.Adding up the area of the small square and the area of the 4 congruent triangles is the area of the big square, the steps are:

c² = (a - b)² + 2ab

= a² - 2ab + b² + 2ab

= a² + b²

Therefore, we get that a² + b² = c² and this is the Pythagoras' Theorem.

Proof #2

We have a trapezium as following:

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There are two ways of counting the area of this trapezium.

The first way is to multiply the height and base with the height of the trapezium then dividing it by 2:

Then evaluate the statement and we get a² + b² + 2ab.

The second way is adding the area of the 6 triangles inside the trapezium as the diagram shown on the top:

+ +

Evaluate the statement and we get 2ab + c².

Equating the two statements again and we get the Pythagoras' Theorem a² + b² = c².

(e) The similarities and differences used in (d)

1.Similarities

A) Both of the proofs used the area of triangles to prove the Pythagoras' Theorem.

b) Both methods requires the combination and the rotation of triangles in order to make the shape at last.

c) Both proofs need to count the area of the shape (e.g. the big square of proof 1 and the trapezium in proof 2).

D) Both of the proofs won't give the identity a² + b² = c² at first but requires further calculations to find the Pythagoras' Theorem.

2.Differences

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a) There are 4 congruent triangles in proof 1 but there are 6 triangles in proof 2.

b) We have to rotate the triangles in proof 1 first but we just need to calculate the area of trapezium first for proof 2.

c) We just have to calculate the area of the big square in proof 1 to get a² + b² = c² but we need to use identity to calculate a² + b² = c² in proof 2.

d) Proof 1 is calculating the area of a square but proof 2 is calculating the area of a trapezium.

(f) Usages of Pythagoras’ Theorem in daily life

1. Earthquake location

Earthquakes result in two different types of waves. When tracking earthquake, geologist can measure the faster and slower waves. By triangulating the distance travelling by both waves, they can determine the center of the earthquake.

2. Arrow, bullet or missile trajectory

By using the Pythagorean Theorem, shooters or archers can calculate the path of the bullet or arrow required to hit the target. If done correctly, the bullet or arrow will hit. If not, it may either fall short or miss. Other projectiles such as guided missiles uses the same method so that they can hit the target accurately.

As shown above, the bullet will descend because of the gravitation pull of earth and its mass. In order to make an accurate shot, the shooter, especially snipers, must use the Pythagorean Theorem to measure the angle of the gun pointer in order make the bullet hit the target.

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As we have mentioned before, the bullet will descend because of gravity. The above diagram shows that the pointer of the gun should be aiming slightly above the line of sight.

(g) Conclusion

A ‘theorem’ is something that has yet to be proved which has not been confirmed. Pythagoras stated his famous theorem: ‘In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.’ Not one of any right angled triangles fails to conform to Pythagoras' Theorem. Therefore the Pythagoras’ Theorem can be argued as the ‘Pythagoras’ formula’ instead of the ‘Pythagoras’ Theorem.

Nowadays, the Pythagoras’ Theorem is a famous and useful mathematical concept. It is one of the most famous Pythagorean proposition. This theorem has made outstanding contributions to the foundation in mathematics. Many real world applications such as construction, navigation and earthquake location rely on the Pythagoras’ Theorem.

(h) Self-reflection

Curtis Baldwin (3):After doing this project, I learnt a lot more about Pythagoras’ Theory. It also taught me the importance of teamwork and time management. This project can't be done if we didn't allocate each of us our own task and manage our time well. I really appreciate that Miss Yeung gave three of us a great opportunity to learn more about Pythagoras Theory, as well as teamwork and time management. Thank you, Miss Yeung, for giving us this chance.

Hugo Ng (9):From this project, I have realized that problems can be solved in many different ways. There are various ways to prove or use Pythagoras’ Theorem in calculations. This inspired me that I have to be flexible and think out of the box. We may achieve the same goal by different methods. We have to choose the method which suits us or is most convenient.

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Humphrey Pang (10): At first when I received this project, I didn't even know what the Pythagoras’ Theorem is, and of course think that it is impossible to prove an identity that has been already proven by someone before. But after doing this project, I then realized how useful Pythagoras' Theorem not just for solving math questions but also for detecting earthquakes which can really save lives also learnt that Mathematician can also save lives, not just solving problems as I thought before. It is quite fun which I had not expected when I tried to discover the proofs for the Pythagoras’ Theorem, there are actually many. But I chose the only two that is the most convincing and easy to understand. I learnt many important things in this project.

(i) Reference

1. http://www.ehow.com/info_8247514_real-life-uses-pythagorean-theorem.html2. http://www.mathsisfun.com/pythagoras.html3. http://www.cut-the-knot.org/pythagoras/index.shtml

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