Given side lengths of a triangle $a, b, c$, judge if this is a acute or obtuse triangle.
One idea came into my mind is using cosine formula, but I wonder if we can do this without using trigonometry. Thank you.
Given side lengths of a triangle $a, b, c$, judge if this is a acute or obtuse triangle.
One idea came into my mind is using cosine formula, but I wonder if we can do this without using trigonometry. Thank you.
The border case between acute and obtuse is right, and a right triangle satisfies the Pythagorean theorem. Thus, denoting the two shorter sides by $a$ and $b$ and the longest one by $c$, the criterion is
$a^2+b^2\lessgtr c^2\;.$
Let $\,a,b,c\,$ be the triangle's sides' lengths. If there exists a relation of the form
$a^2>b^2+c^2$
then you know the angle in front of the side with length $\,a\,$ has to be greater than $\,90^\circ\,$...can you see why?