Pythagorean Theorem

1. Pythagorean Theorem by Bobby Stecher mark.stecher@maconstate.edu

2. c a b The Pythagorean Theorem as some students see it. a2+b2=c2

3. A better way c2 c a2 a a2+b2=c2 b b2

4. A few observations 1. One of the legs of the right triangle is a multiple of 3. 2. One of the legs of the right triangle is a multiple of 4. 3. One of the three is a multiple of 5. Pythagorean Triples (3,4,5) (5,12,13) (7,24,25) (8,15,17) (9,40,41) (11,60,61) (12,35,37) (13,84,85) (16,63,65)

5. Although the Pythagorean Theorem was not known, the Pythagorean triples were familiar to the Babylonians. (Livio, 28) Pythagorean Triples Babylonians discovered that Pythagorean triples can be constructed using the following method. 1. Choose any two whole numbers p and q. Let q be the smaller number. 2. Compute p2 – q2 , 2pq, and p2 + q2

6. 1. Choose any two whole numbers p and q. Let q be the smaller number. Pythagorean Triples Let p = 2 and q = 1. 2. Compute p2 – q2 , 2pq, and p2 + q2 p2 – q2 = 4 – 1 = 3 2pq = 2(2)(1) = 4 p2 + q2 = 4 + 1 = 5 http://www.cut-the-knot.org/Curriculum/Algebra/PythTripleCalculator.shtml

7. The distance formula. (x1,y1) c = distance b = y2-y1 a = x2-x1 (x2,y2) The Pythagorean Theorem is often easier for students to learn than the distance formula.

8. Proof of the Pythagorean Theorem from Euclid Euclid’s Proposition I.47 from Euclid’s Elements.

9. Line segment CN is perpendicular to AB and segment CM is an altitude of ΔABC. Proof of the Pythagorean Theorem

10. Triangle ΔAHB has base AH and height AC. Proof of the Pythagorean Theorem Area of the triangle ΔAHB is half of the area of the square with the sides AH and AC.

11. Triangle ΔACG has base AG and height AM. Proof of the Pythagorean Theorem Area of the triangle ΔACG is half of the area of the rectangle AMNG.

12. AG is equal to AB because both are sides of the same square. Recall that ΔACG is half of rectangle AMNG and ΔAHBis equal to half of square ACKH. Proof of the Pythagorean Theorem AC is equal to AH because both are sides of the same square. Thus square ACKH is equal to rectangle AMNG. Angle <CAG is equal to <HAB. Both angles are formed by adding the angle <CAB to a right angle. ΔACG is equal ΔAHB by SAS.

13. Triangle Δ MBE has base BE and height BC Proof of the Pythagorean Theorem Triangle Δ MBE is equal to half the area of square BCDE.

14. Triangle Δ CBF has base BF and height BM. Proof of the Pythagorean Theorem Triangle Δ CBF is equal to half the area of rectangle BMNF.

15. BE is equal to BC because both are sides of the same square. Thus square BCDE is equal to rectangle BMNF. Proof of the Pythagorean Theorem BA is equal to BF because both are sides of the same square. Angle <EBA is equal to <CBF. Both angles are formed by adding the angle <ABC to a right angle. ΔABE is equal ΔFBC by SAS.

16. World Wide Web java applet for Euclid’s proof. http://www.ies.co.jp/math/java/geo/pythafv/yhafv.html

17. Proof by former president James Garfield. http://jwilson.coe.uga.edu/emt669/Student.Folders/Huberty.Greg/Pythagorean.html Additional Proofs of the Pythagorean Theorem. More than 70 more proofs. http://www.cut-the-knot.org/pythagoras/

18. A simple hands on proof for students. Step 1: Cut four identical right triangles from a piece of paper. c a b

19. Step 2: Arrange the triangles with the hypotenuse of each forming a square. A simple hands on proof for students. a b Area of large square = (a + b)2 a c Area of each part 4 Triangles = 4 x (ab/2) 1 Red Square = c2 b c (a + b)2 = 2ab +c2 c b a2 + 2ab + b2 = 2ab +c2 c a2 + b2 = c2 a b a

20. Alternate arrangement Area of large square = c2 c Area of each part 4 Triangles = 4 x (ab/2) 1 Purple Square = (a – b)2 b a – b a c (a – b)2 + 2ab +c2 a – b a – b c a2 – 2ab + b2 + 2ab = c2 a – b a2 + b2 = c2 c

21. The converse of the Pythagorean Theorem can be used to categorize triangles. If a2 + b2 = c2, then triangle ABC is a right triangle. If a2 + b2 < c2, then triangle ABC is an obtuse triangle. If a2 + b2 > c2, then triangle ABC is an acute triangle.

22. x2 + y2 = r2 is the equation of a circle with the center at origin. Cartesian equation of a circle.

23. Pythagorean Fractal Tree Students can create a fractal using similar right triangles and squares. Using right triangles to calculate and construct square roots.

24. Was Pythagoras a square? The sum of the area’s of the two semi circles on each leg equal to the area of the semi circle on the hypotenuse. The sum of the areas of the equilateral triangles on the legs are equal to the area of the equilateral triangle on the hypotenuse.

25. Extensions and Ideas for lessons Does the theorem work for all similar polygons? Is there a trapezoidal version of the Pythagorean Theorem? Using puzzles to prove the Pythagorean Theorem. Make Pythagorean trees. Cut out triangles and glue to poster board to demonstrate a proof of Pythagorean Theorem. Create a list of Pythagorean triples and apply proofs to specific triples. Use Pythagorean Theorem with the special right triangles. Categorize triangles with converse theorem.

26. Boyer, Carl B. and Merzbach, Uta C. A History of Mathematics 2nd ed. New York: John Wiley & Sons, 1968. Burger, Edward B. and Starbird, Michael. Coincidences, Chaos, and All That Math Jazz. New York: W.W. Norton & Company, 2005. Gullberg, Jan. Mathematics: From the Birth of Numbers. New York: W.W. Norton & Company, 1997. Livio, Mario. The Golden Ratio. New York: Broadway Books, 2002. http://www.contracosta.edu/math/pythagoras.htm http://www.cut-the-knot.org/ http://www.contracosta.edu/math/pythagoras.htm References

27. Links • http://www.contracosta.edu/math/pythagoras.htm • http://www.cut-the-knot.org/ • http://www.contracosta.edu/math/pythagoras.htm • http://www.ies.co.jp/math/java/geo/pythafv/yhafv.html • http://jwilson.coe.uga.edu/emt669/Student.Folders/Huberty.Greg/Pythagorean.html • http://www.cut-the-knot.org/pythagoras/