Question

In part E, you proved that any Pythagorean triple can be generated using the identity (x² − y²)² + (2xy)² = (x² + y²)². Find the missing x- and y-values and Pythagorean triples using the identity given. Write the triple in parentheses, without spaces between the values, with a comma between values, and in order from least to greatest.

a) (3,4,5)
b) (5,12,13)
c) (7,24,25)
d) (8,15,17)

Answer 1

To find missing x- and y-values for the given Pythagorean triples, use the identity (x² - y²)² + (2xy)² = (x² + y²)². For the triples given, the x and y values are calculated, such as x=2, y=1 for the (3,4,5) triple.

Step-by-step explanation:

The student is asking to find the missing x- and y-values that correspond to given Pythagorean triples. Using the identity (x² − y²)² + (2xy)² = (x² + y²)², we can determine the x and y that produce the triples:

• (3,4,5): x = 2, y = 1, as 3 = 2² − 1², 4 = 2*2*1, 5 = 2² + 1²
• (5,12,13): x = 3, y = 2, as 5 = 3² − 2², 12 = 2*3*2, 13 = 3² + 2²
• (7,24,25): x = 4, y = 3, as 7 = 4² − 3², 24 = 2*4*3, 25 = 4² + 3²
• (8,15,17): x = 4, y = 1, as 8 = 4² − 1², 15 = 2*4*1, 17 = 4² + 1²

These x and y values when plugged into the identity produce the corresponding Pythagorean triple.