Finding Pythagorean Triples


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Finding Pythagorean Triples

Student Outcomes

  • Students explore the difference of two squares identity x2 − y2 = (x − y)(x + y) in the context of finding Pythagorean triples.

New York State Common Core Math Algebra II, Module 1, Lesson 10

Worksheets for Algebra II, Module 1, Lesson 10

Classwork

Opening Exercise

Sam and Jill decide to explore a city. Both begin their walk from the same starting point.

  • Sam walks 1 block north, 1 block east, 3 blocks north, and 3 blocks west.
  • Jill walks 4 blocks south, 1 block west, 1 block north, and 4 blocks east.
    If all city blocks are the same length, who is the farthest distance from the starting point?

Example 1:

Prove that if 𝑥 > 1, then a triangle with side lengths 𝑥2 − 1, 2𝑥, and 𝑥2 + 1 is a right triangle

Example 2

Next we describe an easy way to find Pythagorean triples using the expressions from Example 1. Look at the multiplication table below for {1, 2, … , 9}. Notice that the square numbers {1, 4, 9, … , 81} lie on the diagonal of this table.
a. What value of 𝑥 is used to generate the Pythagorean triple (15,8,17) by the formula (𝑥2 − 1, 2𝑥, 𝑥2 + 1)? How do the numbers (1, 4, 4, 16) at the corners of the shaded square in the table relate to the values 15, 8, and 17?
b. Now you try one. Form a square on the multiplication table below whose left-top corner is the 1 (as in the example above) and whose bottom-right corner is a square number. Use the sums or differences of the numbers at the vertices of your square to form a Pythagorean triple. Check that the triple you generate is a Pythagorean triple
Let’s generalize this square to any square in the multiplication table where two opposite vertices of the square are square numbers.
c. How can you use the sums or differences of the numbers at the vertices of the shaded square to get a triple (16, 30, 34)? Is this a Pythagorean triple?
d. Using 𝑥 instead of 5 and 𝑦 instead of 3 in your calculations in part (c), write down a formula for generating Pythagorean triples in terms of 𝑥 and 𝑦.

Relevant Facts and Vocabulary

PYTHAGOREAN THEOREM: If a right triangle has legs of length 𝑎 and 𝑏 units and hypotenuse of length 𝑐 units, then 𝑎2 + 𝑏2 = 𝑐2.
CONVERSE TO THE PYTHAGOREAN THEOREM: If the lengths 𝑎, 𝑏, 𝑐 of the sides of a triangle are related by 𝑎2 + 𝑏2 = 𝑐2, then the angle opposite the side of length 𝑐 is a right angle.
PYTHAGOREAN TRIPLE: A Pythagorean triple is a triplet of positive integers (𝑎, 𝑏, 𝑐) such that 𝑎2 + 𝑏2 = 𝑐2. The triplet (3, 4, 5) is a Pythagorean triple but (1, 1, √2) is not, even though the numbers are side lengths of an isosceles right triangle.




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