Question

which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle? 8, 10, 14 9, 12, 15 10, 14, 17 12, 15, 19

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}

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

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

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

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

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

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

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

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

Answer:

8, 10, 14