Common Pythagorean Triples
Certain sets of three whole numbers satisfy exactly. Memorizing the key families saves time and avoids calculator approximations.
Introduction
A Pythagorean triple is a set of three positive integers where . Recognizing these triples lets you instantly find missing sides without any computation. Any multiple of a triple is also a triple.
Past Knowledge
Pythagorean Theorem (8.1.1). Perfect squares. Factors & multiples.
Today's Goal
Memorize common Pythagorean triples and recognize their multiples.
Future Success
Converse (8.1.3), special right triangles (8.2.1–8.2.2), contest math.
Key Concepts
The Must-Know Triples
| Primitive Triple | Verification | Common Multiples |
|---|---|---|
| 3, 4, 5 | ✓ | 6-8-10, 9-12-15, 12-16-20, 15-20-25 |
| 5, 12, 13 | ✓ | 10-24-26, 15-36-39 |
| 8, 15, 17 | ✓ | 16-30-34 |
| 7, 24, 25 | ✓ | 14-48-50 |
Scaling Property
If is a triple, then is also a triple for any positive integer . This works because .
Worked Examples
Recognizing a Scaled Triple
A right triangle has legs 9 and 12. Find the hypotenuse without calculating.
, → this is
Hypotenuse:
Identifying the Family
A right triangle has a leg of 20 and hypotenuse 25. Find the other leg.
, → this is
Missing leg:
Not a Triple
A right triangle has legs 4 and 7. Is there a Pythagorean triple?
is irrational (65 = 5 × 13, not a perfect square). This is NOT a Pythagorean triple.
Not a triple.
Common Pitfalls
Assuming All Right Triangles Have Integer Sides
Most right triangles do NOT have integer sides. Triples are special cases. Always be prepared for irrational answers like .
Wrong Scaling
When scaling, ALL three numbers must be multiplied by the same factor. You can't scale just two sides of a triple.
Real-Life Applications
Ancient Construction
The Egyptians used a 3-4-5 knotted rope to create perfect right angles when building pyramids. Babylonian tablets from 1800 BCE list Pythagorean triples including 4961-6480-8161!
Screen Resolutions
Common screen resolutions like 1920×1080 don't form perfect Pythagorean triples, but 640×480 (a 4:3 ratio) scales from 3-4-5: the diagonal is pixels.
Practice Quiz
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