Pythagorean Triples - Math Is Fun

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Math is Fun Pythagorean Triples

A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

a2 + b2 = c2

Example: The smallest Pythagorean Triple is 3, 4 and 5.

Let's check it:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

Yes, it is a Pythagorean Triple!

Triangles

Pythagoras

When a triangle's sides are a Pythagorean Triple it is a right angled triangle.

pythagoras squares a^2 + b^2 = c^2

See Pythagoras' Theorem for more details.

Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:

triangle 3 4 5

triangle 3 4 5 lego

Here are two more Pythagorean Triples:

5, 12, 13 9, 40, 41
5, 12, 13 Triangle 9, 40, 41 Triangle
52 + 122 = 132 92 + 402 = 412
25 + 144 = 169 (try it yourself)

And each triangle has a right angle!

List of the First Few

Here is a list of the first few Pythagorean Triples (not including "scaled up" versions mentioned below):

(3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17)
(9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(15,112,113) (16, 63, 65) (17,144,145) (19,180,181)
(20, 21, 29) (20, 99,101) (21,220,221) (23,264,265)
(24,143,145) (25,312,313) (27,364,365) (28, 45, 53)
(28,195,197) (29,420,421) (31,480,481) (32,255,257)
(33, 56, 65) (33,544,545) (35,612,613) (36, 77, 85)
(36,323,325) (37,684,685) ... infinitely many more ...

Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples.

Example: scale 3, 4, 5 by 2 gives 6, 8, 10

Which also fits the formula a2 + b2 = c2:

62 + 82 = 102

36 + 64 = 100

If you want to know more about them read Pythagorean Triples - Advanced

Useful

300, 400 and 500 cm lines

We can use these triples to make a right angle in the real world (such as with carpentry, tiling, etc)

The simple (3,4,5 triple) is the easiest to remember.

And if you need a triple with two nearly equal sides use (119,120,169) or (696,697,985).

619,3252,777,3253,778,620,2242,2243,2244,3254 Pythagoras Theorem Right Angled Triangles Pythagorean Triples - Advanced Triangles Proof that a Triangle has 180° Copyright © 2024 Rod Pierce

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