Done By: 1. Heng Teng Hiang (5) 2. Lee Wen Sin (9) 3. Janice Soo (19) Mentor Group: M07102 (1)Uses of Quadratic Equations (2)Pythagorean Theorem (a) How was it discovered? (b) What are the proofs? (c) What are the numbers that could be a, b and c? (d) How do we use it in our daily life?

Question 1 Why do we learn quadratic equation for? First use: Imagine that you are a policeman. There was a road traffic accident. You know a car’s initial speed, the distance travelled and the car’s acceleration so how are you going to find out the speed of the car? In order to find the speed, we need to find the time taken for the whole journey. Hence, we take the time taken as the unknown quantity. With that, you will be able to form quadratic equations and other linear equations. After you solve the equation (using the simultaneous equation method), you will be able to find speed (distance/time). This will assist you (policeman) to determine other important factors to the accident like: 1) Whether the driver was speeding (breaking the speed limit) 2) Whether the driver is driving with due care 3) The speed of the car when the driver starts to brake 4) The braking time (by finding the road’s coefficient of friction and by measuring the length of the skid mark of the vehicles) With these information, you (a policeman) would be able to decide whether the driver is guilty and if he is to what extent. This will determine his amount of fine and his imprisonment time. Second use: In Europe, paper are measured in A sizes: A0 = 1m 2 A1 = A0/2 A2 = A1/2 A3 = A2/2 A4 = A3/2 Imagine if we had a A3 paper with sides of y (breadth) and x (length). An A4 paper would have the dimension of y and x/2 or 2y and x. If we want the 2 dimension to be equal, it will result into a quadratic equation! So, if we weren’t able to solve quadratic equation, there will be no paper in the world! x y x/2

Question 2 What is it? This theorem which establishes the relationship between the sides of a right-angled triangle was the Pythagorean Theorem. This theorem showed us the relationship between the sides of a right-angled triangle: The square of the hypotenuse = Sum of the squares on the other 2 sides Example: a c b a 2 + b 2 = c 2 (a) How was it discovered? It was discovered by Pythagoras in around 600 B.C. and therefore named after him. Pythagoras was very intelligent and he was hungry for knowledge. He had studied under the Greek named Thales who was discovering the concepts and theory of Geometry. Thales encouraged him to learn everything by himself all around the world at that time. Hence, Pythagoras travelled to Babylon and studied with the Chaldean stargazers. After that, he went to Egypt to study the lore of the priests there. He also studied with the “rope-stretchers” in Egypt. They were actually the engineers who built the pyramids. They knew that if they had a rope with a 12 evenly spaced knots and they pegged them down with at the dimensions of 3, 4, and 5, a right-triangle actually will form. This helped the engineers to build the foundation of the pyramid accurately. However, they kept it a secret, only to themselves. Pythagoras grew fascinated about this rope-stretcher triangle so he spent years trying to find the “magic” behind this. One day, as he was trying to figure out the “magic” while drawing it on the sand, he drew squares “behind” each sides and realized that the sum of the two other shorter sides is equal to the longest side. He named the longest side hypotenuse and this theory or “magic” was named after him as the Pythagorean’s Theorem.

(b) What are the proofs? Although Pythagoras described it as “magic”, it did not just appear or happen to be like that. There are many proofs to show that this theorem is correct. Here’s one proof: 1.Imagine there are 4 right-angled triangles with the same dimensions. (a, b and the hypotenuse: c) 2.Rotate 3 of them to 90 o, 180 o and 270 o. When you put the 4 triangles together, it would form a square. c b a c b a Area of the hole-square: (a-b) 2 Area of the 4 triangles: 4(bh/2) =4(ab/2) = 2ab Area of the whole square: c 2 c 2 = (a-b) 2 + 2ab = a 2 - 2ab + b 2 + 2ab = a 2 + b 2 So c 2 = a 2 + b 2 (c) What are the numbers that could be a, b and c? If we apply it to a set of 3 numbers and it works, the 3 numbers are called the Pythagorean Triple Example: 3, 4 and = = 25 So 3,4 and 5 are Pythagorean Triple. The set could be as small as 20, 21 and 29, it could also be as big as 119, 120 and 169! Pythagorean Triple is infinite.

Sources: (d) How do we use it in our daily life? Pythagorean Theorem has many uses but one of its importance is the measuring of mountains. We cannot just directly measure the mountain but we need it to estimate the danger of the mountain (the composition of air etc.) What should we do? Use Pythagorean Theorem to estimate! Taking this triangle as the mountain… 1) Find out the distance travelled according to the other people’s experience to estimate the distance of the hypotenuse 2) Find the distance of the base and half it. Hypotenuse 2 = height 2 + (base/2) 2 Height 2 = Hypotenuse 2 - (base/2) 2 To estimate the height, you would just need to square root the result! (refer to the model) Height hypotenuse