For how many integers x does a triangle with side lengths 10,29 and x have all its angles acute. how do I know which side is the hypotenuse or does it not matter? Please give me an explanation.

Answer 1

the law of cosines says that if x is the longest side and θ is the opposite angle,

x^2 = 10^2 + 29^2 - 2*10*29 cosθ

If x is less than 10^2+29^2, all three angles are acute.

Of course, if x is not the longest side, then rearrange the numbers a bit to find other conditions.