Question

Problem 1) What are Pythagorean Triples? Please provide a short description below. Use your own words.(5 points)
0 / 10000 Word Limit
Problem 2) How can we prove that three numbers are a Pythagorean Triple? What formula should we use to prove it? Explain.(3 points)
0 / 10000 Word Limit
Problem 3) What shape can we create with Pythagorean Triples? You may provide an example of a Pythagorean Triple to justify your answer.(2 points)

Answer 1

Problem 1) What are Pythagorean Triples?
- Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the relationship between the sides of a right triangle: a^2 + b^2 = c^2. In other words, the two smaller numbers are legs and the largest is the hypotenuse of an integer-sided right triangle.

Problem 2) How can we prove that three numbers are a Pythagorean Triple? What formula should we use to prove it? Explain.
- To prove three numbers a, b, c form a Pythagorean triple, check that a^2 + b^2 = c^2 (with c the largest). If the equality holds and a, b, c are integers, they are a Pythagorean triple.
- (Optional generation fact) All primitive triples (with no common divisor >1) can be produced by integers m>n: a = m^2 − n^2, b = 2mn, c = m^2 + n^2 (with m and n coprime and of opposite parity).

Problem 3) What shape can we create with Pythagorean Triples? Provide an example.
- A right triangle with integer side lengths. Example: the triple (3, 4, 5) gives a right triangle because 3^2 + 4^2 = 9 + 16 = 25 = 5^2.