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 condition for an obtuse triangle

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

Step2: Check option \(8,10,14\)

Let \(a = 8\), \(b=10\), \(c = 14\)
Calculate \(a^{2}+b^{2}=8^{2}+10^{2}=64 + 100=164\)
Calculate \(c^{2}=14^{2}=196\)
Since \(164<196\) (\(a^{2}+b^{2}

Step3: Check option \(9,12,15\)

Let \(a = 9\), \(b = 12\), \(c=15\)
Calculate \(a^{2}+b^{2}=9^{2}+12^{2}=81 + 144=225\)
Calculate \(c^{2}=15^{2}=225\)
Since \(a^{2}+b^{2}=c^{2}\), this is a right - triangle.

Step4: Check option \(10,14,17\)

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

Step5: Check option \(12,15,19\)

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

Answer:

\(8,10,14\)