Problem Solving

AAPL wrote:Which of the following could be the sides of an obtuse angle triangle?

I. 1, 2, 3
II. 2, 3, 4
III. 2, 3, 5

A. I and III only
B. II only
C. III only
D. I and II only
E. I, II and III

The OA is B.

I don't have clear this PS question, why II only? I appreciate if any expert explain it for me. Thank you so much.

In order to be side lengths of a triangle, the triangle inequality tells us that the sum of the two shortest sides is greater than the longest side. In this case, we see that only II (2, 3, 4) are the side lengths of a triangle. Since no answer choice is "None" or "None of these," we can safely say B must be the correct answer without going further to show that the 3 side lengths are indeed the sides of an obtuse angle triangle.

Note: While we have already determined that the answer is B, if we really want to verify that the triangle with sides 2, 3, and 4 is an obtuse triangle, we need the following fact: If c is the longest side in a triangle and a^2 + b^2 < c^2, then the triangle is an obtuse triangle. Since the numbers given to us satisfy 2^2 + 3^2 < 4^2, this triangle is indeed an obtuse triangle.

Answer: B