Question

which sets of three of numbers represent the sides of an obtuse triangle? check all that apply. 4, 7, 8 3, 4, 5 2, 2, 3 6, 8, 9 3, 5, 6

Explanation:

Step1: Recall the obtuse - triangle inequality

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

Step2: Check the set \(\{3,5,6\}\)

Let \(a = 3\), \(b = 5\), \(c = 6\). Calculate \(a^{2}+b^{2}=3^{2}+5^{2}=9 + 25=34\) and \(c^{2}=6^{2}=36\). Since \(34<36\) and \(3 + 5>6\), \(3+6>5\), \(5 + 6>3\), it represents an obtuse - triangle.

Step3: Check the set \(\{6,8,9\}\)

Let \(a = 6\), \(b = 8\), \(c = 9\). Calculate \(a^{2}+b^{2}=6^{2}+8^{2}=36 + 64 = 100\) and \(c^{2}=9^{2}=81\). Since \(100>81\), it does not represent an obtuse - triangle.

Step4: Check the set \(\{2,2,3\}\)

Let \(a = 2\), \(b = 2\), \(c = 3\). Calculate \(a^{2}+b^{2}=2^{2}+2^{2}=4 + 4 = 8\) and \(c^{2}=3^{2}=9\). Since \(8<9\) and \(2 + 2>3\), \(2+3>2\), \(2 + 3>2\), it represents an obtuse - triangle.

Step5: Check the set \(\{3,4,5\}\)

Let \(a = 3\), \(b = 4\), \(c = 5\). Calculate \(a^{2}+b^{2}=3^{2}+4^{2}=9 + 16 = 25\) and \(c^{2}=5^{2}=25\). Since \(a^{2}+b^{2}=c^{2}\), it represents a right - triangle, not an obtuse - triangle.

Step6: Check the set \(\{4,7,8\}\)

Let \(a = 4\), \(b = 7\), \(c = 8\). Calculate \(a^{2}+b^{2}=4^{2}+7^{2}=16+49 = 65\) and \(c^{2}=8^{2}=64\). Since \(65>64\), it does not represent an obtuse - triangle.

Answer:

3,5,6; 2,2,3