Pythagorean slide
- 1. CSO: M.O.8.4.3Objective:Students will solve right triangle problems where the existence of triangles is not obvious using the Pythagorean Theorem.
- 2. Legs – The sides that form the Right (90⁰) angle.Hypotenuse – The side opposite the right angle, it is the longest side of the triangle.Converse – reversing the parts.Helpful Vocabulary
- 3. Pythagorean TheoremDescribes the relationship between the lengths of the legs and the hypotenuse for any right triangleHypotenuseLeg 1Leg 2
- 4. IN WORDS AND SYMBOLSIn a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length c2 = a2 + b2
- 6. Born in Samos (Island in Aegean Sea)Around 570 - 495 BC
- 7. Greek Philosopher, mathematician, and Mystic
- 9. ProofsFirst Video ProofSecond Video ProofThird Video Proof
- 10. Historical NoteWhile we call it Pythagoras‘ Theorem, it was alsoknown by Indian, Greek, Chinese and babylonian mathematicians well before he lived !
- 11. Using a Centimeter Grid to find areaArea = 1 cmArea = 16 cm squaredArea = 48 cm squared
- 12. 3-4-5 RuleThis rule is used to check for the existence of a Right corner.Simply Stated:The measure of any side of 3 units, plus the next side of 4 units has to have a diagonal side of 5 units.
- 13. 3-4-5 Rule ExpandedThis is the 3-4-5 Rule3 squared is 94 squared is 169+16 = 25Square Root of 25 is 5Make a ConjectureIf the length of one side is 6 andLength of the next side is 8,What would be the length of the longest side if this was a Right Triangle and 6 and 8 were the two shorter sides?10
- 14. The answer is 15 since we will not have a negative side to the triangle
- 15. Now let’s try a problem together
- 16. cSide a =12 ftSide b = 18 ftFind the length of the hypotenuse of the above Right Triangle?
- 17. Start withc2 = a2 + b2
- 18. Fill in with knownsc2 = a2 + b2c2 = (12)2 + (18)2
- 19. Square the sidesc2 = 144 + 324Addc2 = 468
- 20. Find the Square Root of Both Sides√c2 = √468Round c = 21.63
- 21. If you reverse the parts of the pythagorean theorem, you have formed itsConverse, and it is also true
- 22. Funny Break
- 24. Irrational numbers and PythagorasAn irrational number is a number that cannot be expressed as the quotient a/b where a and b are integers and b ≠ 0Every square root of an imperfect square is an irrational number.Example:√10 = 3.1622776……..This number continues indefinitely with no repetition
- 25. Problems to tryc2= a2 + b2c2 = 24yds2 + 18yds2c2 = 576 + 324c2 = 900c = 30b2 = c2 - a2b2 = 82 - 32b2 = 64 – 9b2 = 55b= 7.42a2 = c2- b2a2 = 20cm2 - 17cm2a2 = 400 - 289a2 = 111a = 10.54
- 26. Answer to this problem using Pythagoras is 8ft
- 27. 22 ft14 ftHow tall does the ladder need to be to reach the coconuts?
- 28. Hope you learned something about Pythagoras and his theorem.
- 29. ReferencesWho2 Biography. Copyright © 1998-2010 by Who2, LLC. All rights reserved. See the Pythagoras biography from Who2. Pierce, Rod. "Math is Fun - Maths Resources" Math Is Fun. Ed. Rod Pierce. 19 Apr 2010. 1 Oct 2010 http://www.mathsisfun.com/ http://www.glencoe.com/ose/showbook.php