Triangle $ABC$ has vertices $A(-1, 1, 3)$, $B(-1, 3, 5)$, and $C(-3, 3, 3)$.
What kind of triangle is $ABC$? Justify your answer.
So far all I have done is I found the distance between $AB$, $BC$ AND $AC$, I do not know whether I should use either the Pythagorean theorem, cosine law or both?
The cosine law is just a generalization of Pythagorean theorem.
By just exploiting the Pythagorean theorem we have:
$$ a^2 =BC^2 = 2^2+0^2+2^2 = 8, $$ $$ b^2 = AC^2 = 2^2+2^2+0^2=8, $$ $$ c^2 = AB^2 = 0^2+2^2+2^2=8, $$ and $a=b=c$ gives that your triangle is equilateral.
If you have a bit of visualization skills, you may immediately see that your triangle
is something like the shaded triangle here:
$\hspace{2in}$