1. CSO: M.O.8.4.3 Objective: 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 Theorem Describes the relationship between the lengths of the legs and the hypotenuse for any right triangle Hypotenuse Leg 1 Leg 2

4. IN WORDS AND SYMBOLS In 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

5. Who is Pythagoras?

6. Born in Samos (Island in Aegean Sea)Around 570 - 495 BC

7. Greek Philosopher, mathematician, and Mystic

9. Proofs First Video Proof Second Video Proof Third Video Proof

10. Historical Note While we call it Pythagoras‘ Theorem, it was also known by Indian, Greek, Chinese and babylonian mathematicians well before he lived !

11. Using a Centimeter Grid to find area Area = 1 cm Area = 16 cm squared Area = 48 cm squared

12. 3-4-5 Rule This 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 Expanded This is the 3-4-5 Rule 3 squared is 9 4 squared is 16 9+16 = 25 Square Root of 25 is 5 Make a Conjecture If the length of one side is 6 and Length 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. c Side a =12 ft Side b = 18 ft Find the length of the hypotenuse of the above Right Triangle?

17. Start with c2 = a2 + b2

18. Fill in with knowns c2 = a2 + b2 c2 = (12)2 + (18)2

19. Square the sides c2 = 144 + 324 Add c2 = 468

20. Find the Square Root of Both Sides √c2 = √468 Round c = 21.63

21. If you reverse the parts of the pythagorean theorem, you have formed its Converse, and it is also true

22. Funny Break

23. Pythagorean Triples

24. Irrational numbers and Pythagoras An irrational number is a number that cannot be expressed as the quotient a/b where a and b are integers and b ≠ 0 Every square root of an imperfect square is an irrational number. Example: √10 = 3.1622776…….. This number continues indefinitely with no repetition

25. Problems to try c2= a2 + b2 c2 = 24yds2 + 18yds2 c2 = 576 + 324 c2 = 900 c = 30 b2 = c2 - a2 b2 = 82 - 32 b2 = 64 – 9 b2 = 55 b= 7.42 a2 = c2- b2 a2 = 20cm2 - 17cm2 a2 = 400 - 289 a2 = 111 a = 10.54

26. Answer to this problem using Pythagoras is 8ft

27. 22 ft 14 ft How tall does the ladder need to be to reach the coconuts?

28. Hope you learned something about Pythagoras and his theorem.

29. References Who2 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