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I'm teaching a geometry class and want to ensure my students understand the most common errors and misconceptions related to the Pythagorean Theorem and its applications.

I attempted an initial Google search, but it didn't produce any significant results.

Any references would be appreciated.

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    $\begingroup$ I don't know any off-hand (other than the Freshman's dream), but I have often USED the Pythagorean theorem to argue for the existence of an error, namely that $\sqrt{x+y} = \sqrt x + \sqrt y$ is NOT true (in general). If it were true, then $\sqrt{a^2 + b^2} = a+b$ for a triangle with side lengths $a$ & $b$ and hypothenuse length $c,$ and hence the Pythagorean theorem violates the fact that the shortest distance between two points must be along the line joining the two points. $\endgroup$ Oct 4, 2023 at 14:59
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    $\begingroup$ It wasn't discovered by Pythagoras, but I don't think that's what you're looking for. $\endgroup$ Oct 5, 2023 at 14:36
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    $\begingroup$ Very famously, the scarecrow in the Wizard of Oz movie (1939) misquotes the Pythagorean theorem. $\endgroup$ Oct 5, 2023 at 17:43

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Here's a list of mistakes that I've seen students make.

Conceptual Mistakes

  • Applying the Pythagorean Theorem on non-right triangles. (They may also think that the word "hypotenuse" means the longest side of any triangle.)

  • Assuming that the missing side is always the square root of the sum of squares of the other sides -- even when that missing side is a leg, not the hypotenuse. (They think the unknown side is always $c$ in the formula $a^2 + b^2 = c^2.$)

  • Writing their final answer with $\pm$ even though the side length cannot be negative. (They use $c = \pm \sqrt{a^2 + b^2}$ because they're used to solving quadratic equations with no domain constraints, e.g., $x^2 = 9 \Rightarrow x = \pm 3.$)

Mindless Mistakes in Algebra

  • Forgetting to take the square root at the end (i.e., $c = a^2 + b^2$).

  • Forgetting to square the quantities within the square root (i.e., $c = \sqrt{a + b}$).

  • Using a product instead of addition (i.e., $c = \sqrt{ab}$).

Misconceptions About Algebra

  • Distributing the square root (i.e., $c = \sqrt{a^2 + b^2}$ $= \sqrt{a^2} + \sqrt{b^2}$ $= a + b$).

  • "Canceling out" exponents of all terms in an equation (i.e., $a^2 + b^2 = c^2$ $\Rightarrow a + b = c$). Note that while this is technically the same error as distributing the square root, most students who make this error won't understand that and will continue to make it even if they understand that they cannot distribute a square root.

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    $\begingroup$ Mixing sides up so common, it helps not to make exercises consist of a block of "find the hypotenuse" questions then a block of "find the missing legs". Varying questions forces students to assess each situation from scratch, as needed in an exam. One "cure" for beginners: always draw a diagram, with squares sticking out from each side, and write their areas inside. It's clear which 2 small squares sum to the bigger. Also consider questions with multiple answers: "Two sides of a right-angled triangle are 4 cm and 5 cm. What values can the area of the triangle be? Draw a diagram for each case." $\endgroup$
    – Silverfish
    Oct 5, 2023 at 12:04
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    $\begingroup$ One more suggestion to this fine list: some high school syllabuses, e.g. England's GCSE, extend things slightly to distinguish between acute, right-angled and obtuse triangles when all three sides are given, depending on whether $c^2$ (where $c$ is the longest side) is less than, equal to, or greater than $a^2 + b^2$; or similarly whether an angle is acute, right or obtuse (the slight difference being that $c$ is now the side opposite the angle of interest). Mixing up which inequality implies acute versus obtuse is a common error. $\endgroup$
    – Silverfish
    Oct 5, 2023 at 12:13
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Reversing the implication: if $a^2+b^2 = c^2$, then a,b,c are the side-lengths of a right triangle. With the right restrictions that's true, but it's not what the Pythagorean theorem claims.

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Comment from JosephDoggie on the question:

Very famously, the scarecrow in the Wizard of Oz movie ($1939$) misquotes the Pythagorean Theorem.

Looking up the quote on Google gives this resource which gives this bit of info:

At the end of The Wizard of Oz, the Scarecrow receives a diploma and then immediately says, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side."

Looking up "Isosceles triangle" on Google gives a calculator and switching what we need to solve for from "base" to "side" gives us $a=\frac P2-\frac b2$ (where $P=2a+b$) because an isosceles triangle has two sides that are equal to each other. The statement made by the Scarecrow would only be true if and only if $b=0$ (while $a$ could be equal to anything) because we have$$\sqrt a+\sqrt b=\sqrt a\implies b=0,a=c_1$$where $c_1$ is any number.

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    $\begingroup$ It seems the Scarecrow was ahead of his time. That sounds like something an AI would say nowadays. $\endgroup$ Oct 6, 2023 at 2:02
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    $\begingroup$ Homer Simpson recites the Scarecrow quote too: youtube.com/watch?v=iDqFw40eIY0 $\endgroup$
    – Nayuki
    Oct 6, 2023 at 6:46

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