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PYTHAGORAS'. THEOREM. {4} The Improving Mathematics Education in Schools (TIMES) Project. ASSUMED KNOWLEDGE. • Familiarity with measurement of lengths, angles and area. • Basic knowledge of congruence and similarity. • Familiarity with simple geometric proofs. • Simple geometric constructions.
Problem 10: Find an algorithm that determines whether a given Diophantine (i.e., multi-variable polynomial) equation has any integer solutions. Ex: x2+y2=z2 has many integer solutions. (Pythagorean theorem, e.g., x="3", y="4", z="5") x9+y9=z9 has no integer solutions (corollary of Fermat?s. Last Theorem, conjectured in 1637,
121 proofs of the Pythagorean theorem: squares on the legs of a right triangle add up to the square on the hypotenuse.
Investigate the history of Pythagoras and the Pythagorean Theorem. Also, have the opportunity to practice applying the Pythagorean Theorem to several problems. Students should analyze information on the Pythagorean Theorem including not only the meaning and application of the theorem, but also the proofs.
2012 MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved. Proofs of the Pythagorean Theorem MATHEMATICAL GOALS This lesson unit is
2] Four Thousand Years of Discovery. 27. 2.1] Pythagoras and the First Proof. 27. 2.2] Euclid's Wonderful Windmill. 36. 2.3] Liu Hui Packs the Squares. 45. 2.4] Kurrah Transforms the Bride's Chair. 49. 2.5] Bhaskara Unleashes the Power of Algebra. 53. 2.6] Leonardo da Vinci's Magnificent Symmetry 55. 2.7] Legendre
Introduction. The Theorem of Pythagoras is a well-known theorem. It is also a very old one, not only does it bear the name of Pythagoras, an ancient Greek, but it was also known to the ancient Babylonians and to the ancient Egyptians. Most school students learn of it as a2 + b2 = c2. The actual statement of the theorem is
Pythagorean Theorem Proofs. G. M. Wysin, wysin@phys.ksu.edu, www.phys.ksu.edu/personal/wysin. Proof # 1. Inscribe objects inside the c2 square, and add up their areas. b s a f f f f f f f f f f f f f f f c a f f f f f f f f f f f f f f f s b c. Total area = hypothenuse squared = c2. Inscribed area = four triangles + one square. What is s?
Attempt 1: Suppose a right triangle has sides of length a, b and c. Draw squares on the three sides as shown. Divide these squares into smaller squares. You can see that the number of squares on the two shorter sides add up to make the number of squares on the longest side. So: a2+b2=c2 a b c a b c. Attempt 2. Suppose
Page 1. Proof of Pythagoras theorem a b c a b c a b c a b c. (a + b)2 = 4. (ab. 2. ) + c2 a2 +P. PP. 2ab + b2 = P. PP. 2ab + c2 a2 + b2 = c2.
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