Question: Am I missing something here? My homework says to prove that the given triangle is a right triangle, but it does not appear to be a right triangle mathematically.

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> Let $A =(-3, 2)$, $B=(1, 0)$, and $C=(4,6)$.
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> Prove that $\blacktriangle ABC$ is a right triangle. 
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I have tried the slope method and it didn't work, I also tried the distance (Pythagorean) method and that didn't work either.

Answer: 
Let's try the Pythagorean theorem:

Side $AB = [1- -3,0-2] = [4,-2]$, with length $\sqrt{20}$

Side $BC = [4- 1,6-0] = [3,6]$, with length $\sqrt{45}$

Side $AC = [4- -3,6-2] = [7,4]$, with length $\sqrt{65}$

$|AB|^2 + |BC|^2 = (\sqrt{20})^2+(\sqrt{45})^2 = 20 + 45 = 65 = (\sqrt{65})^2 = |AC|^2$

So the triangle must be right
Translate the triangle by $(-1,0)$. The new coordinates of the vertices become:
$$ A(-4,2),\quad B(0,0),\quad C(3,6) $$
and since the dot product between $A$ and $C$ is zero, $\widehat{ABC}$ is a right angle.