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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.
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.
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.
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.
