The converse of the Pythagorean theorem

What is the inverse of the Pythagorean theorem used for?

The Pythagorean theorem allows us to calculate a side length when we ALREADY KNOW that a triangle is right-angled. The inverse does the opposite: it allows us to CHECK whether a triangle is right-angled, using the lengths of its three sides, without knowing the angles.

The statement of the converse

If, in a triangle, the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is right-angled (and the right angle is opposite the longest side).

Method of verification

1. Identify the longest side (potential hypotenuse)
2. Calculate (longest side)²
3. Calculate the sum of the squares of the other two sides
4. Compare the two results: if they are equal, the triangle is a right-angled triangle; otherwise, it is not

Example: a right-angled triangle

Triangle with sides of 5 cm, 12 cm and 13 cm. The longest side is 13. 13² = 169 5² + 12² = 25 + 144 = 169 The two results are equal: the triangle is a right-angled triangle.

Example: a triangle that is not a right-angled triangle

A triangle with sides of 5 cm, 6 cm and 8 cm. The longest side is 8. 8² = 64 5² + 6² = 25 + 36 = 61 64 is not equal to 61: the triangle is NOT a right-angled triangle.

Pitfall to avoid

To apply the inverse, you must always square the LARGEST of the three sides, not a side chosen at random. Choosing the wrong side completely distorts the comparison and may lead to the incorrect conclusion that a triangle is not right-angled, or the opposite.