Pythagorean triples are sets of three positive whole numbers that can be the side lengths of a right triangle. They matter because they let you recognize right triangles and find missing sides quickly without using decimals. The most famous example is 3, 4, 5 because 3^2 + 4^2 = 5^2.
In geometry, triples save time on diagrams, coordinate problems, construction, and distance calculations.
Key Facts
- A Pythagorean triple is three positive integers a, b, and c such that a^2 + b^2 = c^2.
- The hypotenuse c is always the longest side and is always across from the right angle.
- Common triples include 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41.
- Multiplying a triple by the same positive integer makes a new triple, such as 3-4-5 becoming 6-8-10.
- Primitive triples have no common factor greater than 1, such as 3-4-5 and 5-12-13.
- Euclid's formula generates triples: a = m^2 - n^2, b = 2mn, c = m^2 + n^2, where m > n > 0.
Vocabulary
- Pythagorean triple
- A set of three positive integers that satisfies a^2 + b^2 = c^2 and can form a right triangle.
- Hypotenuse
- The longest side of a right triangle, located opposite the right angle.
- Leg
- One of the two shorter sides of a right triangle that meet to form the right angle.
- Primitive triple
- A Pythagorean triple whose three numbers have no common factor greater than 1.
- Scale factor
- A number used to multiply every side length of a figure or triple to make a similar larger or smaller version.
Common Mistakes to Avoid
- Putting the largest number in a or b instead of c. The largest side must be the hypotenuse because c^2 is the sum of the squares of the two legs.
- Adding side lengths instead of squaring them. The rule is a^2 + b^2 = c^2, not a + b = c.
- Multiplying only one or two numbers in a triple to make a new triple. A scaled triple works only when all three numbers are multiplied by the same scale factor.
- Assuming any three numbers close to a known triple must be a right triangle. You must check the equation, such as 6^2 + 8^2 = 10^2, before using the shortcut.
Practice Questions
- 1 A right triangle has legs 9 and 12. Find the hypotenuse and identify whether the side lengths form a scaled Pythagorean triple.
- 2 Determine whether 10, 24, and 26 form a Pythagorean triple. Show the square calculation that proves your answer.
- 3 A triangle has side lengths 7, 24, and 25. Explain how you can recognize quickly that it is a right triangle and identify the hypotenuse.