Acute or Obtuse?

When the test does not give zero, its sign still tells you something. Add the squares of the two shorter sides, then subtract the square of the longest side. If the result is positive, the largest angle is less than 90° and the triangle is acute. If the result is negative, the longest side is longer than it would be in a right triangle. The largest angle is then more than 90°, and the triangle is obtuse.

Sides 4, 5 and 6 give a positive result. Sides 5, 6 and 9 give a negative one.

Sides 4, 5 and 6: 16 + 25 - 36 = 5. The result is positive, so the triangle is acute. Sides 5, 6 and 9: 25 + 36 - 81 = -20. The result is negative, so the triangle is obtuse. The figure shows why. When the longest side gets longer, the angle facing it opens wider. Once the square of the longest side is bigger than the other two squares added together, that angle is more than 90°.