Pythagorean triples


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Pythagorean triples

  1. 1. Pythagorean Triples<br />Big Idea:<br />Pythagorean triple is any set of three whole numbers that satisfies the Pythagorean theorem c2 = a2 + b2. A primitive Pythagorean triple has the additional characteristic that the greatest common divisor of a, b, and c is 1.<br />Goal: <br />Understand and recognize primitive Pythagorean triples and their elementary properties; use Euclids’s formula to generate primitive Pythagorean triples; prove Euclid’s formula.<br />
  2. 2. Pythagorean Triples<br />Any set of three whole numbers that satisfies the Pythagorean<br />theorem is called a Pythagorean triple. Examples of Pythagorean triples include {3, 4, 5}, {6, 8, 10}, and {5, 12, 13}. (Pythagorean Triples are generally denoted in brackets of the form {a,b,c}.<br />Notice that {3, 4, 5} and {6, 8, 10} represent similar triangles. The sides of similar right triangles are proportional to each other. So multiples of Pythagorean triples are also Pythagorean triples. <br />For any given Pythagorean triple there exists an infinite number of triples of the form k{a,b,c}, where k is a constant.<br />10<br />5<br />3<br />6<br />4<br />8<br />
  3. 3. Pythagorean Triples<br />Whole number triples whose greatest common divisor is 1 are considered primitive. Thus {3, 4, 5} is a primitive Pythagorean triple and {6, 8, 10} is not.<br />Integers a and b whose greatest common factor is 1 are said to be coprime or relatively prime. For example: 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3. <br />
  4. 4. Pythagorean Triples<br />Let’s try to find additional primitive Pythagorean triples. In other words, find a set of positive integers a, b, and c such that a 2 + b 2 = c 2where a, b, and c are coprime.<br />“Guess & check” is one strategy to find Pythagorean triples, but it is time consuming.<br />Euclid developed a formula for finding such primitives. The formula states that an arbitrary pair of positive integers m and n with m> n will form a Pythagorean triple if m and n are coprime and one of them is odd. The formula is:<br /> a = m2 – n2<br /> b = 2mn<br /> c = m2 + n2.<br />
  5. 5. Pythagorean Triples<br />Activity PT #1 <br />Use Euclid’s formula to find three additional primitive Pythagorean triples other than {3,4,5} and {5,12,13}. <br />Activity PT #2 <br />Use basic algebra to prove Euclid’s formula that c2 = a2 + b2 when <br />a = m2 – n2, b = 2mn and c = m2 + n2.<br />
  6. 6. Pythagorean Triples<br />Elementary properties of primitive Pythagorean triples<br />(c-a)(c-b)/2 is always a perfect square.<br />Exactly one of a, b is odd; c is odd.<br />Exactly one of a, b is divisible by 3.<br />Exactly one of a, b is divisible by 4.<br />Exactly one of a, b, c is divisible by 5.<br />Exactly one of a, b, (a + b), (b – a) is divisible by 7.<br />Exactly one of (a + c), (b + c), (c – a), (c – b) is divisible by 8.<br />Exactly one of (a + c), (b + c), (c – a), (c – b) is divisible by 9.<br />Exactly one of a, b, (2a + b), (2a – b), (2b + a), (2b – a) is divisible by 11.<br />The hypotenuse exceeds the even leg by the square of an odd integer j, and exceeds the odd leg by twice the square of an integer k>0, from which it follows that: There are no primitive Pythagorean triples in which the hypotenuse and a leg differ by a prime number greater than 2.<br />
  7. 7. Pythagorean Triples<br />Activity PT #3 <br />Select five of the elementary properties of primitive Pythagorean triples and prove that they hold true for the three primitive Pythagorean triples you created in Activity #1.<br />
  8. 8. Pythagorean Triples<br />A bit of history…<br />Around 4000 years ago, the Babylonians and the Chinese used the concept of the Pythagorean triple {3, 4, 5} to construct a right triangle by dividing a long string into twelve equal parts, such that one side of the triangle is three, the second side is four and the third side is five sections long. <br />In India (8th - 2nd century BC), the BaudhayanaSulba Sutra contained a list of Pythagorean triples, a statement of the theorem and the geometrical proof of the theorem for an isosceles right triangle.<br />Pythagoras (569–475 BC), used algebraic methods to construct Pythagorean triples. He was not universally credited with this for another 500 years.<br />The ancient Greek philosopher Plato (c. 380 BC) used the expressions 2n, n2 – 1, and n2 + 1 to produce Pythagorean triples.<br />