which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle?\n8, 10, 14\n9…

Explanation:

Step1: Recall the obtuse - triangle inequality

For a triangle with side lengths (a), (b), and (c) ((c) being the longest side), the triangle is obtuse if (a^{2}+b^{2}<c^{2}).

Step2: Check option 1 ((a = 8), (b = 10), (c = 14))

Calculate (a^{2}+b^{2}) and (c^{2}). (a^{2}+b^{2}=8^{2}+10^{2}=64 + 100=164), (c^{2}=14^{2}=196). Since (164<196), this set of side - lengths can form an obtuse triangle.

Step3: Check option 2 ((a = 9), (b = 12), (c = 15))

Calculate (a^{2}+b^{2}=9^{2}+12^{2}=81 + 144 = 225), (c^{2}=15^{2}=225). Since (a^{2}+b^{2}=c^{2}), this is a right - triangle, not an obtuse triangle.

Step4: Check option 3 ((a = 10), (b = 14), (c = 17))

Calculate (a^{2}+b^{2}=10^{2}+14^{2}=100+196 = 296), (c^{2}=17^{2}=289). Since (296>289), this is an acute - triangle.

Step5: Check option 4 ((a = 12), (b = 15), (c = 19))

Calculate (a^{2}+b^{2}=12^{2}+15^{2}=144 + 225=369), (c^{2}=19^{2}=361). Since (369>361), this is an acute - triangle.

Answer:

8, 10, 14