The triangle with side lengths 11, 12, and 15 is an obtuse triangle. Hence the correct option is b.
In a triangle, the nature of the angles depends on the relationship between the square of the lengths of its sides. For an obtuse triangle, the square of the length of the longest side (the hypotenuse) is greater than the sum of the squares of the lengths of the other two sides.
In the given triangle with side lengths 11, 12, and 15, we can apply the Pythagorean theorem to check for the nature of the angles:
c^2 >a^2 +b^2
Where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
For this triangle:
15^2 >11^2 +12^2
225>121+144
225>265
The inequality holds true, indicating that the triangle is obtuse. Therefore, the given lengths (11, 12, 15) form an obtuse triangle.