What exactly is the converse and the reciprocal of the Pythagorean theorem?

Question

I think I spotted a translation problem at exercise 12 in Appendix B of Stewart's Calculus, 6th edition, the Portuguese version of the book. Here's the original problem:

12. (a) Show that the triangle with vertices $A(6, -7)$, $B(11, -3)$ and $C(2, -2)$ is a right triangle using the converse of the Pythagorean Theorem. (b) Use slopes to show that $ABC$ is a right triangle. (c) Find the area of the triangle.

The translation however puts it this way: 12. (a) Show that the triangle with vertices $A(6, -7)$, $B(11, -3)$ and $C(2, -2)$ is a right triangle using the reciprocal of the Pythagorean Theorem.

These are two different theorems, aren't they? The converse should be --- if the square of the hypotenuse equals the sum of the squares of the other sides, then there's a right angle in the triangle. The reciprocal theorem says that $1/h^2 = 1/a^2 + 1/b^2$ where $h$ is the height of the triangle relative to the right-angle.

Or am I not quite understanding the essence of the exercise? Please advise.

The Portuguese version in Portuguese. "12. (a) Mostre que o triângulo com vértices $A(6, -7)$, $B(11, -3)$ e $C(2, -2)$ é um triângulo retângulo usando a recíproca do Teorema de Pitágoras. (b) Use as inclinações para mostrar que $ABC$ é um triângulo retângulo. (c) Determine a área do triângulo."

Here's an image.

Answer 1

If we go to the Portuguese Wikipedia article on the Pythagorean theorem, we see the following:

A recíproca do teorema de Pitágoras também é verdadeira: "Para qualquer triângulo com lados $l$, $m$, e $r$, se $l^2 + m^2 = r^2$, então o ângulo entre $l$ e $m$ mede $90^\circ$".

This tells us that the statement of "recíproca do teorema de Pitágoras" is what in English we call the converse to the Pythagorean theorem: if a triangle with side lengths $l$, $m$, and $r$ satisfies $l^2 + m^2 = r^2$, then the angle between the sides of lengths $l$ and $m$ is a right angle.

There is no mistake in the translation, and the use of "recíproca" has no relation to the English "reciprocal", nor to the theorem you are quoting as the "reciprocal theorem".

Note that in non-mathematical language, the word "reciprocal" refers to something done in return for a previous action, or something done in both directions. In English, the mathematical usage ended up with using "reciprocal" for the multiplicative inverse, and it appears that in Portuguese, the mathematical usage ended up with using the corresponding word for the converse, but it is easy to see how both are related to the original meaning.

In Portuguese, the multiplicative inverse appears to just be called inverso multiplicativo, if we are to trust Wikipedia a second time. The word "recíproco" is only mentioned once, as a historical usage in a translation from English.