# Does it follow from the Pythagorean theorem that "there is no right-angled triangle with sides 5, 10 and 11"?

Assume, for the sake of this question, that we define the Pythagoeran theorem as

Theorem 1: In a right-angled triangle, the sum of the squares of the two shortest sides is equal to the square of the longest side.

Then, this theorem's converse would be

Theorem 2: If the sum of the squares of the two shortest sides is equal to the square of the longest side, the triangle is right-angled.

To make the distinction between these two theorems clearer, consider this statement which does follow from the second theorem but not from Pythagoras':

"A triangle with side lengths $$3$$, $$4$$ and $$5$$ is right-angled."

As for my question, from which of the two theorems does the below statement follow?

"There is no right triangle with sides $$5$$, $$10$$ and $$11$$.".

• It follows from the contrapositive of the first. A triangle whose sides are such that the sum of the squares of the smaller two sides is not equal to the square of the larger side is necessarily not a right triangle. The contrapositive of an implication $P\implies Q$ is logically equivalent to the original implication and reads instead as $\neg Q\implies \neg P$ Commented May 3, 2021 at 19:50
• But wouldn't the statement also follow from the contrapositive of the second theorem, which would be along the lines of "If the triangle isn't right-angled then the sum of the squares of the two shortest sides is inequal to the square of the longest side"`? Commented May 3, 2021 at 20:00
• No. Make sure you understand the difference between $P\implies Q$ and $Q\implies P$. These are totally different statements which are in most scenarios not the same and can have different truth values. Here, letting $P$ be "Is a right triangle with side lengths $a\leq b\leq c$" and $Q$ be "Is a triangle with side lengths $a\leq b\leq c$ such that $a^2+b^2=c^2$" the first theorem reads $P\implies Q$. The statement you are interested in is $\neg Q\implies \neg P$ which is, as already mentioned, the contrapositive of the first... not the second. Commented May 3, 2021 at 20:06
• That is, "$5^2+10^2\neq 11^2$ so the triangle is not a right triangle." Do not confuse this with $\neg P\implies \neg Q$. That is the "inverse" of the first and that is not logically equivalent to the first, but it is the contrapositive of the second and thus logically equivalent to the second. Commented May 3, 2021 at 20:07
• Why is the statement "There is no right triangle with sides 5, 10 and 11." equivalent to "$5^2+10^2≠11^2$ so the triangle is not a right triangle."? Also, if you present your comments as an answer to this post, I will accept it as one. Commented May 3, 2021 at 20:16

## 1 Answer

Theorem 1 is the Pythagorean theorem. Theorem 2 is its converse (not its contrapositive).

Both theorems are true.

The fact that there is no right triangle with sides $$5$$, $$10$$ and $$11$$ follows from the contrapositive of Theorem 1, so from Theorem 1. It's a different way of stating the same implication.