Pythagorean Triangle/Examples/3-4-5
Example of Primitive Pythagorean Triangle
The triangle whose sides are of length $3$, $4$ and $5$ is a primitive Pythagorean triangle.
Proof
| \(\ds 3^2 + 4^2\) | \(=\) | \(\ds 9 + 16\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 25\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5^2\) |
It follows by Pythagoras's Theorem that $3$, $4$ and $5$ form a Pythagorean triple.
Note that $3$ and $4$ are coprime.
Hence, by definition, $3$, $4$ and $5$ form a primitive Pythagorean triple.
The result follows by definition of a primitive Pythagorean triangle.
$\blacksquare$
Also see
- Smallest Pythagorean Triangle is 3-4-5
- Pythagorean Triangle whose Area is Half Perimeter: its area and semiperimeter are both $6$
- Pythagorean Triangle with Sides in Arithmetic Sequence
Historical Note
To the Pythagoreans, the had particular significance: the sides of lengths $3$ and $4$ denoted the male and female principles, while the hypotenuse of length $5$ denoted their offspring.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6$
- 1990: William Dunham: Journey Through Genius ... (previous) ... (next): Chapter $1$: Hippocrates' Quadrature of the Lune ($\text {ca. 440}$ b.c.)
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Pythagorean Triples
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$