## 1. Proof of Pythagoras Theorem

### 1.1. METHOD 1

1.1.1. Suppose that you start with four right triangles with sides of length a, b and c and a square tray with sides of length a+b

1.1.2. Arrange the triangles in two different ways. In the first way, you leave two square holes. These have a combined area of a2 + b2. In the second way, you leave one large square hole. This has an area of c2. Since these areas are equal, a2 + b2 = c2

### 1.2. METHOD 2

1.2.1. Now we start with four copies of the same triangle. Three of these have been rotated 90°, 180°, and 270°, respectively. Each has area ab/2. Let's put them together without additional rotations so that they form a square with side c.

1.2.2. The square has a square hole with the side (a - b). Summing up its area (a - b)² and 2ab, the area of the four triangles (4·ab/2), we get c² = (a - b)² + 2ab c² = a² - 2ab + b² + 2ab c² = a² + b²

## 2. History of Pythagoras

### 2.1. A famous Ionian (Greek) philosopher and mathematician who is know for coming up with the Pythagoras Theorem.

### 2.2. born during 570 BC in Samos

### 2.3. travelled extensively and studied under different teachers

### 2.4. opened a school called the Semicircle

### 2.5. his disciples were called Pythagoreans

### 2.6. had many discoveries in different fields(scientific, mathematical, music, etc.)

### 2.7. even though the theorem was already discovered by the Babylonians, Pythagoras was the first to prove it

### 2.8. he was married to Theano, his first pupil at Croton, who was also a philosopher in her own right. however some said she was not his wife but merely disciple.

### 2.9. had four children, but some sources say he had up to seven children

## 3. What is the Pythagorean Theorem

### 3.1. It is a fundamental relation among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

## 4. Converse

### 4.1. If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

## 5. Real-Life Applications of Pythagoras Theorem

### 5.1. Going on a Roadtrip

5.1.1. Let’s say two friends are meeting at a playground. Mary is already at the park but her friend Bob needs to get there taking the shortest path possible. Bob has two way he can go - he can follow the roads getting to the park - first heading south 3 miles, then heading west four miles. The total distance covered following the roads will be 7 miles. The other way he can get there is by cutting through some open fields and walk directly to the park. If we apply Pythagoras's theorem to calculate the distance you will get: (3)2 + (4)2 = 9 + 16 = C2 √25 = C 5 Miles. = C

### 5.2. Painting a Wall

5.2.1. Painters use ladders to paint on high buildings and often use the help of Pythagoras' theorem to complete their work. The painter needs to determine how tall a ladder needs to be in order to safely place the base away from the wall so it won't tip over. In this case the ladder itself will be the hypotenuse. Take for example a painter who has to paint a wall which is about 3 m high. The painter has to put the base of the ladder 2 m away from the wall to ensure it won't tip. What will be the length of the ladder required by the painter to complete his work? You can calculate it using Pythagoras' theorem: (5)2 + (2)2 = 25 + 4 = C2 √100 = C 5.3 m. = C Thus, the painter will need a ladder about 5 meters high.