1) The document discusses Pythagoras' theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 2) It provides two proofs of the theorem - one using the areas of four rotated triangles to form a square, and one using the areas of triangles in a trapezoid. 3) Applications of the theorem discussed include determining earthquake locations and calculating projectile trajectories like bullets or arrows.

1. 1
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

2. 2
(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

3. 3
(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:

4. 4
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:
(2a + 2b)
2 x (a + b)
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:
2a x b
2
+
2b x a
2
+
2 x c2
2
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.

5. 5
2.Differences
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.

6. 6
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.

7. 7
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.html
2. http://www.mathsisfun.com/pythagoras.html
3. http://www.cut-the-knot.org/pythagoras/index.shtml