The Converse & Pythagorean Triples

The Pythagorean theorem tells you a side when you already know the triangle is right. Its converse runs the other way: if a² + b² = c², the triangle *must* be right.

That turns it into a classification test. With c as the longest side:

• a² + b² = c² → right
• a² + b² > c² → acute
• a² + b² < c² → obtuse

For 6, 8, 11: 36 + 64 = 100, which is less than 121, so the triangle is obtuse.

Pythagorean triples are whole-number side sets. Knowing them on sight saves real time: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41.

Any multiple of a triple is also a triple, so 6-8-10 and 30-40-50 are both right triangles.

The converse states that if a² + b² = c² with c the longest side, the angle opposite c is right. It is proved by constructing a right triangle with legs a and b, applying the theorem to get hypotenuse c, and concluding congruence by SSS.

The comparison version generalises this into a classification and follows from the Law of Cosines, c² = a² + b² − 2ab·cos C: the sign of a² + b² − c² matches the sign of cos C, which is positive for acute C, zero for right and negative for obtuse.

A necessary first check is the triangle inequality — 3, 4, 20 satisfies no triangle at all, so classifying it is meaningless.

Primitive Pythagorean triples are those with gcd(a,b,c) = 1, and all are generated by a = m² − n², b = 2mn, c = m² + n² for coprime m > n of opposite parity. Scaling any triple by an integer produces another, which is why recognising 9-12-15 as 3×(3,4,5) is faster than computing.