Which set of numbers represents the side lengths, in millimeters, of an obtuse triangle? A. 8, 10, 14 B. 9, 12, 15 C. 10, 14, 17 D. 12, 15, 19

To determine whether a set of numbers can represent the side lengths of an obtuse triangle, you need to assess whether one of the angles in the triangle is obtuse, that is, greater than 90 degrees.

For a triangle with sides a, b, and c, where c is the longest side, the triangle is obtuse if \( a^2 + b^2 < c^2 \). This is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Let's now test each set of numbers:

A. 8, 10, 14 Using the Pythagorean inequality for obtuse triangles: \( 8^2 + 10^2 < 14^2 \) 64 + 100 < 196 164 < 196

Since 164 is less than 196, triangle A could be obtuse.

B. 9, 12, 15 \( 9^2 + 12^2 < 15^2 \) 81 + 144 < 225 225 < 225

Since 225 is not less than 225, triangle B cannot be obtuse. This one actually represents a right triangle because \( 9^2 + 12^2 = 15^2 \).

C. 10, 14, 17 \( 10^2 + 14^2 < 17^2 \) 100 + 196 < 289 296 < 289

Since 296 is not less than 289, triangle C cannot be obtuse.

D. 12, 15, 19 \( 12^2 + 15^2 < 19^2 \) 144 + 225 < 361 369 < 361

Since 369 is not less than 361, triangle D cannot be obtuse.

So, out of the options given, only set A (8, 10, 14) can represent the side lengths of an obtuse triangle.