## Pythagoras' Theorem

by ELIZABETH CHUA KAI XIAN 2S
## 1. Pythagoras Triples/Triplets

### 1.1. 3,4,5 3^2 + 4^2 = 5^2 9 + 16 = 25

### 1.2. 5,12,13 5^2 + 12^2 = 13^ 25 + 144 = 169

### 1.3. A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: a^2 + b^2 = c^2

## 2. Delegations

### 2.1. Liz: History of Pythagoras

### 2.2. Tiffany: What is it?

### 2.3. Shannon: Proof

### 2.4. Joelle: Converse of Pythagoras’ Theorem

### 2.5. 4 in total, everyone do one: Applications

## 3. What is it?

### 3.1. When a triangle is right-angled and triangles are formed on all three sides of it, the largest square has the same area as the two other squares combined.

3.1.1. The Pythagoras' Theorem can be written as one equation: a^2 + b^2 = c^2

## 4. Converse of Pythagoras Theorem

### 4.1. If a^2+b^2=c^2 holds

4.1.1. Then DABC is a right angled triangle with right angle at C

### 4.2. How to prove the converse of pythagoras theorem

4.2.1. EF = BC = a ÐF is a right angle. FD = CA = b In DDEF, By Pythagoras Theorem, ……..(2) By (1), the given, Theorefore, AB = DE But by construction, BC = EF and CA= FD

## 5. History of Pythagoras

### 5.1. The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.?-500 B.C.?), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras.

5.1.1. EXAMPLE ONE: Ancient clay tablets from Babylonia indicate that the Babylonians in the second millennium B.C., 1000 years before Pythagoras, had rules for generating Pythagorean triples, understood the relationship between the sides of a right triangle, and, in solving for the hypoteneuse of an isosceles right triangle, came up with an approximation of accurate to five decimal places. [They needed to do so because the lengths would represent some multiple of the formula: 12 + 12 = ()2.]

5.1.1.1. EXAMPLE TWO: Chinese astronomical and mathematical treatise, Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C.), possibly predating Pythagoras, gave a statement of and geometrical demonstration of the Pythagorean theorem

5.1.1.1.1. EXAMPLE THREE: Indian mathematicians, in the ancient times, also knew the Pythagorean theorem, and the Sulbasutras (of which the earliest date from ca. 800-600 B.C.) discuss it in the context of strict requirements for the orientation, shape, and area of altars for religious purposes. It has also been suggested that the ancient Mayas used variations of Pythagorean triples in their Long Count calendar.

### 5.2. Pythagoras himself was not simply a mathematician. He was an important philosopher who believed that the world was ruled by harmony and that numerical relationships could best express this harmony. He was the first, for example, to represent musical harmonies as simple ratios. Pythagoras and his followers were also a bit eccentric. Pythagoras's followers were sworn to absolute secrecy, and their devotion to their master bordered on the cult-like. Pythagoreans followed a strict moral and ethical code, which included vegetarianism because of their belief in the reincarnation of souls. They also refused to eat beans!

## 6. Proof

### 6.1. Proof using Algebra

6.1.1. Area of Whole Square It is a big square, with each side having a length of a+b, so the total area is: A = (a+b)(a+b)

6.1.2. Area of The Pieces Now let's add up the areas of all the smaller pieces: First, the smaller (tilted) square has an area of c^2 And there are four triangles, each one has an area of ½ab So all four of them combined is A = 4(½ab) = 2ab So, adding up the tilted square and the 4 triangles gives c^2+2ab

6.1.3. Both Areas Must Be Equal The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as: (a+b)(a+b) = c2+2ab NOW, let us rearrange this to see if we can get the pythagoras theorem: Start with: (a+b)(a+b) = c^2 + 2ab Expand (a+b)(a+b): a^2 + 2ab + b^2 = c2 + 2ab Subtract "2ab" from both sides: a^2 + b^2 = c^2