Proof, Converse and Pythagorean Triples

1Why this matters

Checking 3, 4, 5 and a few other triangles does not prove that a² + b² = c² for every right-angled triangle. A proof does. Its converse is just as useful: Rwandan builders lay out foundations with a rope marked 3 m, 4 m and 5 m (or 6 m, 8 m and 10 m), because a triangle with those sides must have a right angle.

2A proof by rearrangement

Take any right-angled triangle with legs a, b and hypotenuse c. Make two large squares, each of side a + b, and fill them in two different ways.

• Square 1 holds a square of area a², a square of area b², and two a-by-b rectangles. Each rectangle is two copies of the triangle, so Square 1 holds a² + b² + four triangles.
• Square 2 holds four copies of the triangle in its corners, leaving a tilted square in the middle whose side is the hypotenuse c. So Square 2 holds c² + four triangles.

The two large squares are the same size. Remove the four triangles from each, and what remains must be equal: a² + b² = c². Because a and b were any lengths, the theorem holds for every right-angled triangle.

3The converse

If the sides of a triangle satisfy a² + b² = c², then the angle opposite c is a right angle.

4Pythagorean triples

Three whole numbers a, b, c with a² + b² = c² form a Pythagorean triple. Well-known ones: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).

• Any multiple of a triple is a triple: (6, 8, 10) and (9, 12, 15) come from (3, 4, 5).
• To generate new triples, choose whole numbers m > n and use a = m² − n², b = 2mn, c = m² + n². With m = 2 and n = 1 you get (3, 4, 5); m = 3, n = 2 gives (5, 12, 13); m = 4, n = 1 gives (15, 8, 17).

The builder's rope. To mark a right angle for a foundation, a mason measures 6 m along one string line and 8 m along the other, then moves the second line until the diagonal between the two marks is exactly 10 m. Since 6² + 8² = 10², the corner is a right angle.

Practice