How do you know if a triangle is acute, obtuse, or a right triangle by looking at it's side lengths?

Answer 1

Take the 2 shortest sides, a and b, square each and add them

compare that sum to the square of the longest side, c.

If they're equal, it's a right triangle

and the longest side is the hypotenuse: a² + b² = c²

IF the sum of squares of the 2 shortest sides > the square of the longest side, the triangle is acute

a² + b² > c²

IF the sum of squares of the 2 shortest sides < square of the longest side, it's an obtuse triangle

a² + b² < c²

^{This can be seen from the law of cosines}

c² = a² + b² - 2abcosC

If C=90 degrees, it's a right triangle and the last term is zero, as cos(90) = 0

If C>90, it's an obtuse triangle and cosC<0 making c² < a² + b²

If C<90, it's an acute triangle and cosC>0 making the square of the longest side c less than the sum of squares of the two shorter sides.