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1 CT  PPT  How Zhao Yin  Ho Qi Yan Ong Ru Yun   Pythagoras and the Pythagorean Theorem 2014
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1 CT PPT How Zhao Yin Ho Qi Yan Ong Ru Yun Pythagoras and the Pythagorean Theorem 2014

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  • 1. NAME : HOW ZHAO YIN (201420026) HO QI YAN (201420009) ONG RU YUN (201420030) CLASS : FOUNDATION IN LIBERAL ARTS MODULE : INTRODUCTION TO CRITICAL THINKING LECTURER : MS. DOT MACKENZIE TERM : MAY 2014 DATE : 14 JULY 2014 TOPIC : PYTHAGORAS AND THE PYTHAGOREAN THEOREM
  • 2. PYTHAGORAS AND THE PYTHAGOREAN THEOREM Illustration source: http://www.edb.utexas.edu/visionawards/petrosino/Media/Members/zhfbdzci/pythagoras1.gif According to the UALR (2001), “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.”
  • 3. WHERE WAS PYTHAGORAS BORN? Samo s Illustration source: http://intmstat.com/blog/2008/03/samos.jpg
  • 4. WHAT IS PYTHAGOREAN THEOREM? According to the UALR (2001), “The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypoteneuse, or, in mathematical terms, for the triangle shown at right, a2 + b2 = c2. Integers that satisfy the conditions a2 + b2 = c2 are called "Pythagorean triples." ”
  • 5. RIGHT-ANGLED TRIANGLE Illustration source: http://www.mathopenref.com/images/triangle/hypotenuse.gif
  • 6. Illustration source: http://www.mathsaccelerator.com/measurement/images/triangle-answer.gif Right-Angled Triangle
  • 7. HOW TO PROVE THE EQUATION OF PYTHAGOREAN THEOREM? c b a •There are four similar triangle with the rotation of different angle which are 90°, 180°, and 270°. •Area of triangle can be calculated by using this formulae: ½ x a x b
  • 8. •The four triangles combined together to form a square shape with a square hole. •The length of side of square inside is a-b. •The area of square inside is (a-b)² or 2ab. •The area of four triangles is 4(½ x a x b). In the last, we get this formulae c²= (a - b)² + 2ab = a² - 2ab + b² + 2ab = a² + b²
  • 9. Illustration source: http://cdn.instructables.com/FN4/7VG4/GVZPOZOZ/FN47VG4GVZPOZOZ.MEDIUM.gif
  • 10. Video source: http://www.youtube.com/watch?v=hTxqdyGjtsA&feature=related
  • 11. EXERCISE 1: Prove triangle X is a right-angled triangle. http://fc05.deviantart.net/fs70/f/2013/297/b/1/simple_background_by_biebersays-d6rnj7n.jpg
  • 12. SOLUTIONS: c2= b2+a2 Let AC2=AB2+BC2 AB2+BC2=82+152 AC2=64+225 =289 √AC2=√289 AC=17 cm http://fc05.deviantart.net/fs70/f/2013/297/b/1/simple_background_by_biebersays-d6rnj7n.jpg
  • 13. EXERCISE 2: Assuming that triangle Q is a right-angled triangle, find the length of side YZ. http://hqwide.com/minimalistic-multicolor-gaussian-blur-simple-background-white-wallpaper-5602/
  • 14. SOLUTIONS: c2=b2+a2 Let ZY2=ZX2+XY2 ZX2+XY2=122+52 ZY2=144+25 =169 √ZY2=√169 ZY=13 cm http://hqwide.com/minimalistic-multicolor-gaussian-blur-simple-background-white-wallpaper-5602/
  • 15. (12, 35, 37) http://hqwide.com/gaussian-blur-gradient-simple-background-blurred-colors-wallpaper-62699/
  • 16. 122+352=144+1225 =1369 372=1369 122+352=372, therefore it has been proved that 12, 35, 37 are the sides of a Pythagorean Triangle. http://hqwide.com/gaussian-blur-gradient-simple-background-blurred-colors-wallpaper-62699/
  • 17. http://www.wallsave.com/wallpaper/1920x1080/simple-light-gradient-211999.html
  • 18. c2=b2+a2 52=b2+a2 Let a=b-1, b=a+1 52=b2+(b-1)2 52=b2+(b-1)(b-1) 52=b2+b2-b-b+1 25=2b2-2b+1 ÷2 0=2b2-2b-24 0=b2-b-12 0=(b+3)(b-4) Hypotenuse b+3=0 b=-3 b should be positive, therefore b=-3 is not acceptable. b-4=0 b=4 cm a+1=b a+1=4 a=3 cm Therefore, lengths of legs=3 cm, 4 cm http://www.wallsave.com/wallpaper/1920x1080/simple-light-gradient-211999.html
  • 19. REFERENCES Bogomolny, A. (2012). Pythagorean Theorem. Retrieved July 9, 2014, from Cut The Knot: http://www.cut-the-knot.org/pythagoras/. Section 9.6 The Pythagorean Theorem. (2007). Retrieved July 10, 2014, from 2014, from Msenux Redwoods: http://msenux.redwoods.edu/IntAlgText/chapter9/section6solutions.pdf. Smoller, L. (2001, May). The History of Pythagorean Theorem. Retrieved July 10, 2014, from UALR College of Information Science and Systems Engineering: http://ualr.edu/lasmoller/pythag.html.
  • 20. http://www.ucsa.nl/wp-content/uploads/2012/10/Questions-and-Answers.jpeg