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:

For a triangle with side lengths \(a \leq b \leq c\):

1. First confirm it is a valid triangle: \(a + b > c\)
2. Classify the triangle:

• Obtuse: \(a^2 + b^2 < c^2\)
• Right: \(a^2 + b^2 = c^2\)
• Acute: \(a^2 + b^2 > c^2\)

Step1: Test 4,7,8

Check validity: \(4+7>8\) (valid).
Check obtuse: \(4^2 + 7^2 = 16 + 49 = 65\), \(8^2 = 64\). \(65 > 64\) (acute)

Step2: Test 3,4,5

Check validity: \(3+4>5\) (valid).
Check obtuse: \(3^2 + 4^2 = 9 + 16 = 25\), \(5^2 = 25\). \(25 = 25\) (right)

Step3: Test 2,2,3

Check validity: \(2+2>3\) (valid).
Check obtuse: \(2^2 + 2^2 = 4 + 4 = 8\), \(3^2 = 9\). \(8 < 9\) (obtuse)

Step4: Test 6,8,9

Check validity: \(6+8>9\) (valid).
Check obtuse: \(6^2 + 8^2 = 36 + 64 = 100\), \(9^2 = 81\). \(100 > 81\) (acute)

Step5: Test 3,5,6

Check validity: \(3+5>6\) (valid).
Check obtuse: \(3^2 + 5^2 = 9 + 25 = 34\), \(6^2 = 36\). \(34 < 36\) (obtuse)

Answer:

2, 2, 3
3, 5, 6