Question

which of the following sets of numbers could not represent the three sides of a triangle? answer \\{9,23,31\\} \\{11,15,28\\} \\{14,22,34\\} \\{12,16,26\\}

Explanation:

Step1: Recall Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For a set of three numbers \(a\), \(b\), \(c\) (where \(c\) is the largest), we need \(a + b>c\).

Step2: Check \(\{9, 23, 31\}\)

Largest side is \(31\). Check \(9 + 23\): \(9+23 = 32\). Since \(32>31\), this can be a triangle.

Step3: Check \(\{11, 15, 28\}\)

Largest side is \(28\). Check \(11 + 15\): \(11 + 15=26\). Since \(26<28\), this does not satisfy the triangle inequality.

Step4: Check \(\{14, 22, 34\}\)

Largest side is \(34\). Check \(14 + 22\): \(14+22 = 36\). Since \(36>34\), this can be a triangle.

Step5: Check \(\{12, 16, 26\}\)

Largest side is \(26\). Check \(12 + 16\): \(12+16 = 28\). Since \(28>26\), this can be a triangle.

Answer:

\(\{11, 15, 28\}\) (because \(11 + 15 = 26<28\), violating the triangle inequality theorem)