Final answer:
The student's question relates to Pythagorean triples and Pythagoras' theorem. The given pairs (x1, y1) and (x2, y2) cannot generate Pythagorean triples as they result in irrational numbers when applying the theorem to find the hypotenuse. Pythagorean triples consist of three positive integers that fit the relationship a² + b² = c².
Step-by-step explanation:
The student is asked about calculating Pythagorean triples associated with given pairs of values and determining for which values xn and yn a certain identity is valid. A Pythagorean triple consists of three positive integers (a, b, c) that fit the formula a² + b² = c², where 'c' is the hypotenuse and 'a' and 'b' are the other two sides of a right triangle.
The values given by the student, x1 = 5, y1 = 4, and x2 = 1, y2 = 1/2, do not represent integers and therefore cannot generate integer-based Pythagorean triples as described by Pythagoras' theorem. Pythagorean triples are always integers, and fractions are not allowed.
However, if we interpret the request as finding the length of the hypotenuse when the legs of the right triangle are given, we can use Pythagoras' theorem. Using this theorem described by Pythagoras, we can find the hypotenuse 'c' using the equation c = √(a² + b²).
For the first pair, using x1 = 5 and y1 = 4, the hypotenuse c1 would be calculated as:
c1 = √(5² + 4²) = √(25 + 16) = √41
For the second pair, using x2 = 1 and y2 = 1/2, the hypotenuse c2 would be calculated as:
c2 = √(1² + (1/2)²) = √(1 + 1/4) = √5/2
Note that these results provide irrational numbers and therefore do not form traditional Pythagorean triples. Since a Pythagorean triple must be composed of integers, x1 and y1 can generate a Pythagorean triple if x1 and y1 are integers, and if their squared sum equals the square of another integer.