What are Pythagorean Triples? Definition and Examples

📚 Quick Study Guide

• 📐 Definition: A Pythagorean triple consists of three positive integers $a$, $b$, and $c$, such that $a^2 + b^2 = c^2$.
• ➕ Formula: The fundamental equation to remember is $a^2 + b^2 = c^2$, where $c$ represents the length of the hypotenuse in a right triangle, and $a$ and $b$ are the lengths of the other two sides.
• 🌱 Primitive Triples: A Pythagorean triple is considered primitive if $a$, $b$, and $c$ are coprime (i.e., their greatest common divisor is 1).
• 💡 Generating Triples: Pythagorean triples can be generated using the formulas: $a = m^2 - n^2$, $b = 2mn$, $c = m^2 + n^2$, where $m$ and $n$ are positive integers with $m > n$.
• 📝 Common Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) are frequently encountered Pythagorean triples.
• ➕ Multiples: Any multiple of a Pythagorean triple is also a Pythagorean triple. For instance, since (3, 4, 5) is a triple, so is (6, 8, 10).

Practice Quiz

1. Which of the following sets of numbers is a Pythagorean triple?
   1. (1, 2, 3)
   2. (2, 3, 4)
   3. (3, 4, 5)
   4. (4, 5, 6)
2. Which equation defines a Pythagorean triple ($a$, $b$, $c$)?
   1. $a + b = c$
   2. $a^2 + b^2 = c^2$
   3. $a^3 + b^3 = c^3$
   4. $a - b = c$
3. Is (6, 8, 10) a primitive Pythagorean triple?
   1. Yes
   2. No
   3. Cannot be determined
   4. Only on Tuesdays
4. Which of the following is NOT a Pythagorean triple?
   1. (5, 12, 13)
   2. (8, 15, 17)
   3. (7, 24, 25)
   4. (9, 40, 42)
5. If $a = m^2 - n^2$, $b = 2mn$, and $c = m^2 + n^2$, what are $a$, $b$, and $c$ if $m = 2$ and $n = 1$?
   1. $a = 3$, $b = 4$, $c = 5$
   2. $a = 5$, $b = 3$, $c = 4$
   3. $a = 2$, $b = 3$, $c = 4$
   4. $a = 1$, $b = 2$, $c = 3$
6. Which of these triples could represent the sides of a RIGHT triangle?
   1. (2, 2, 3)
   2. (4, 5, 6)
   3. (6, 8, 10)
   4. (1, 1, 1)
7. If (3, 4, 5) is a Pythagorean triple, which of the following is also a Pythagorean triple?
   1. (4, 5, 6)
   2. (9, 12, 15)
   3. (1, 2, 3)
   4. (2, 3, 4)