Pythagoras Theorem: Pythagoras Theorem By: Abhishek Sehgal, Abobaker Ugool, Neelkanth Bhular
PowerPoint Presentation: This is a right triangle:
PowerPoint Presentation: We call it a right triangle because it contains a right angle.
PowerPoint Presentation: The measure of a right angle is 90 o 90 o
PowerPoint Presentation: The little square 90 o in the angle tells you it is a right angle.
PowerPoint Presentation: The two sides which come together in a right angle are called
PowerPoint Presentation: The two sides which come together in a right angle are called
PowerPoint Presentation: The two sides which come together in a right angle are called legs.
PowerPoint Presentation: The lengths of the legs are usually called a and b . a b
PowerPoint Presentation: The side across from the right angle a b is called the hypotenuse.
PowerPoint Presentation: And the length of the hypotenuse is usually labeled c . a b c
Right angled triangles Summary: Right angled triangles Summary It is a triangle which has a 90 degrees’ angle. The two sides that make up the right angle are called legs . The side opposite the right angle is the hypotenuse . leg leg hypotenuse right angle
PowerPoint Presentation: About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.
PowerPoint Presentation: Pythagorus realized that if you have a right triangle, 3 4 5
PowerPoint Presentation: and you square the lengths of the two sides that make up the right angle, 3 4 5
PowerPoint Presentation: and add them together, 3 4 5
PowerPoint Presentation: you get the same number you would get by squaring the other side. 3 4 5
PowerPoint Presentation: History
PowerPoint Presentation: Who was Pythagoras ? Pythagoras was one of the greatest mathematician ever known. Today he is mainly known for discovering Pythagorean Theorem.
~ Belief ~: Even though it was Pythagoras who got credits for discovering the relationship be right angled triangle but it is now believed that Pythagoras actually got the idea from Egyptians. Therefore, Pythagoras was not the first one to discover the theorem; there were many other people who discovered it before him but did not introduce it in the society. ~ Belief ~
PowerPoint Presentation: Pythagoras was born on 569 B.C in island of Samos, Greece.
Education: Not much known about his formal schooling but he travelled a lot especially through Egypt and gained informal education and learning along with mathematics one of them. His childhood education was under Pherecydes. At age of 18, he went to island of Lesbos. Then he accepted Anaximander as his teacher and studied astronomer and philosopher. Later he went to Egypt ,then at that time Cambyses invaded Egypt and was prisoned and sent to Babylon. There for twelve years he studied with Magi and when he left Babylon he went to India. In India he continued his education under Brachmanes. Later he left India as a teacher. E ducation
PowerPoint Presentation: He made a special cult known as “The brotherhood of Pythagoreans”. They were interested in in philosophy, but especially in music and mathematics. They liked maths because they thought math is the rule of how the world works.
Death: Pythagoras died in 500 - 475 B.C in Metapontum Italy. It is said that he was killed by angry mob because he caught up in a war between the Agrigentum and the Syracusans and killed by the Syracusans and their is also a belief that he was burned out of his school in Crotona and then went to Metapontum where he starved himself to death. There are many stories about how he died but the above ones are most common. But, most of the include a scene where Pythagoras refuses to trample a crop of bean plants in order to escape and because of this, he is caught. Death
PowerPoint Presentation: How does Pythagorean Theorem work? a 2 + b 2 = c 2 This is how most of us learn Pythagorean Theorem a c b But what we did not learn is why does it work and how does it work?
PowerPoint Presentation: Pythagoras found out that for any right angled triangle the area of the squares on the two shorter sides will add together to give the area of the square on the longer side. So the area of the two blue squares will add together to give the area of the yellow square.
PowerPoint Presentation: What is Pythagorean Theorem Used For ? To find the length of the missing side in a right angled triangle. 5 4 n
PowerPoint Presentation: Proof
PowerPoint Presentation: # 1’st Proof The two blue squares will add together to give the area of the yellow square. 4 4 3 3 9 16 9 + 16 = 25 If we know the area of the yellow square we can work out how long one of its sides is by using the square root. √ 25 = 5 Thus it is true for any right angled triangle.
Arranging triangle with the hypotenuse to form a square: Arranging triangle with the hypotenuse to form a square a b c b b a a b c c c a Area of large square = (a + b) 2 Area of each part 4 Triangles = 4 x (ab/2) 1 Red Square = c 2 (a + b) 2 = 2ab +c 2 a 2 + 2ab + b 2 = 2ab +c 2 a 2 + b 2 = c 2 # 2’nd Proof
Alternate arrangement: Alternate arrangement Area of large square = c 2 Area of each part 4 Triangles = 4 x (ab/2) 1 Purple Square = (a – b) 2 (a – b) 2 + 2ab +c 2 a 2 – 2ab + b 2 + 2ab = c 2 a 2 + b 2 = c 2 b c a – b c c c a a – b a – b a – b # 3’rd Proof
PowerPoint Presentation: You can use The Pythagorean Theorem to solve many kinds of problems. Suppose you drive directly west for 48 miles, 48
PowerPoint Presentation: Then turn south and drive for 36 miles. 48 36
PowerPoint Presentation: How far are you from where you started? 48 36 ?
PowerPoint Presentation: 48 2 Using The Pythagorean Theorem, 48 36 c 36 2 + = c 2
PowerPoint Presentation: Why? Can you see that we have a right triangle? 48 36 c 48 2 36 2 + = c 2
PowerPoint Presentation: Which side is the hypotenuse? Which sides are the legs? 48 36 c 48 2 36 2 + = c 2
PowerPoint Presentation: Then all we need to do is calculate:
PowerPoint Presentation: And you end up 60 miles from where you started. 48 36 60 So, since c 2 is 3600, c is 60. So, since c 2 is 3600, c is
PowerPoint Presentation: Find the length of a diagonal of the rectangle: 15" 8" ?
PowerPoint Presentation: Find the length of a diagonal of the rectangle: 15" 8" ? b = 8 a = 15 c
PowerPoint Presentation: b = 8 a = 15 c
PowerPoint Presentation: Find the length of a diagonal of the rectangle: 15" 8" 17
PowerPoint Presentation: Use of Pythogorean Theorem in Real Life Situation : How far up a wall will an 11m ladder reach , if the foot of ladder must be 4m from the base of the wall ?
PowerPoint Presentation: Use of Pythogorean Theorem in Real Life Situation : How far up a wall will an 11m ladder reach , if the foot of ladder must be 4m from the base of the wall ? 11 2 = X 2 +4 2 121 = x 2 + 16 121 - 16 = x 2 105 = x 2 10.2 = x The ladder will reach 10.2 meters up the wall.
PowerPoint Presentation: Problems Faced During This Task
PowerPoint Presentation: Problems Faced During our research accurate information was hard to find and in the part of proofs; there were too many and hard to decide which proof was the best but we overcame it by choosing random and most common information. We could have worked on time management and organization skills and also make the presentation a little bit more better. It would also have been much easier if we had more time as a group and in class. Abobaker faced problems in finding reliable sources and same with Neelkanth. Abhishek found problems in organising and researching proof. We had to change the task assigned and remove some information intended to submit primarily. But, aside from that everything else went ok.
PowerPoint Presentation: Credits Slide Show Abhishek Sehgal & Neelkanth Bhular History Abobaker Ugool Abhishek Sehgal & Neelkanth Bhular Proof Abhishek Sehgal Examples Abobaker Ugool Abhishek Sehgal & Neelkanth Bhular How does it work? Abhishek Sehgal
PowerPoint Presentation: Credits Script Abobaker Ugool Abhishek Sehgal & Neelkanth Bhular Editing Abhishek Sehgal Structure Abobaker Ugool Abhishek Sehgal & Neelkanth Bhular Notes Abobaker Ugool Abhishek Sehgal & Neelkanth Bhular Director Abhishek Sehgal