Pythagorean slide

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Pythagorean slide

  1. 1. CSO: M.O.8.4.3<br />Objective:<br />Students will solve right triangle problems where the existence of triangles is not obvious using the Pythagorean Theorem. <br />
  2. 2. Legs – The sides that form the Right (90⁰) angle.<br />Hypotenuse – The side opposite the right angle, it is the longest side of the triangle.<br />Converse – reversing the parts.<br />Helpful Vocabulary<br />
  3. 3. Pythagorean Theorem<br />Describes the relationship between the lengths of the legs and the hypotenuse for any right triangle<br />Hypotenuse<br />Leg 1<br />Leg 2<br />
  4. 4. IN WORDS AND SYMBOLS<br />In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length <br />c2 = a2 + b2<br />
  5. 5. Who is Pythagoras?<br />
  6. 6. Born in Samos (Island in Aegean Sea)Around 570 - 495 BC<br />
  7. 7. Greek Philosopher, mathematician, and Mystic<br />
  8. 8.
  9. 9. Proofs<br />First Video Proof<br />Second Video Proof<br />Third Video Proof<br />
  10. 10. Historical Note<br />While we call it Pythagoras‘ Theorem, it was also<br />known by Indian, Greek, Chinese and babylonian mathematicians well before he lived ! <br />
  11. 11. Using a Centimeter Grid to find area<br />Area = 1 cm<br />Area = 16 cm squared<br />Area = 48 cm squared<br />
  12. 12. 3-4-5 Rule<br />This rule is used to check for the existence of a Right corner.<br />Simply Stated:<br />The measure of any side of 3 units, plus the next side of 4 units has to have a diagonal side of 5 units. <br />
  13. 13. 3-4-5 Rule Expanded<br />This is the 3-4-5 Rule<br />3 squared is 9<br />4 squared is 16<br />9+16 = 25<br />Square Root of 25 is 5<br />Make a Conjecture<br />If the length of one side is 6 and<br />Length of the next side is 8,<br />What would be the length of the longest side if this was a Right Triangle and 6 and 8 were the two shorter sides?<br />10<br />
  14. 14. The answer is 15 since we will not have a negative side to the triangle<br />
  15. 15. Now let’s try a problem together<br />
  16. 16. c<br />Side a =12 ft<br />Side b = 18 ft<br />Find the length of the hypotenuse of the above Right Triangle? <br />
  17. 17. Start with<br />c2 = a2 + b2<br />
  18. 18. Fill in with knowns<br />c2 = a2 + b2<br />c2 = (12)2 + (18)2<br />
  19. 19. Square the sides<br />c2 = 144 + 324<br />Add<br />c2 = 468<br />
  20. 20. Find the Square Root of Both Sides<br />√c2 = √468<br />Round <br />c = 21.63<br />
  21. 21. If you reverse the parts of <br />the pythagorean theorem, you have formed its<br />Converse, and it is also true<br />
  22. 22. Funny Break<br />
  23. 23. Pythagorean Triples<br />
  24. 24. Irrational numbers and Pythagoras<br />An irrational number is a number that cannot be expressed as the quotient a/b where a and b are integers and b ≠ 0<br />Every square root of an imperfect square is an irrational number.<br />Example:<br />√10 = 3.1622776……..<br />This number continues indefinitely with no repetition <br />
  25. 25. Problems to try<br />c2= a2 + b2<br />c2 = 24yds2 + 18yds2<br />c2 = 576 + 324<br />c2 = 900<br />c = 30<br />b2 = c2 - a2<br />b2 = 82 - 32<br />b2 = 64 – 9<br />b2 = 55<br />b= 7.42<br />a2 = c2- b2<br />a2 = 20cm2 - 17cm2<br />a2 = 400 - 289<br />a2 = 111<br />a = 10.54<br />
  26. 26. Answer to this problem using Pythagoras is 8ft<br />
  27. 27. 22 ft<br />14 ft<br />How tall does the ladder need to be to reach the coconuts?<br />
  28. 28. Hope you learned something about Pythagoras and his theorem. <br />
  29. 29. References<br />Who2 Biography. Copyright © 1998-2010 by Who2, LLC. All rights reserved. See the Pythagoras biography from Who2. <br /> <br /> <br />Pierce, Rod. "Math is Fun - Maths Resources" Math Is Fun. Ed. Rod Pierce. 19 Apr 2010. 1 Oct 2010 http://www.mathsisfun.com/<br /> <br />http://www.glencoe.com/ose/showbook.php<br /> <br />

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