Question

which of the following sets of numbers could represent the three sides of a triangle? answer {14,25,39} {14,16,30} {14,27,40} {11,24,36}

Explanation:

Step1: Recall Triangle Inequality Theorem

For three side lengths \(a\), \(b\), \(c\) (where \(c\) is the largest), the sum of the two smaller sides must be greater than the largest side: \(a + b>c\).

Step2: Check \(\{14,25,39\}\)

Largest side \(c = 39\). Sum of smaller sides: \(14 + 25=39\). But \(39\) is not greater than \(39\) (it's equal), so this is not a triangle.

Step3: Check \(\{14,16,30\}\)

Largest side \(c = 30\). Sum of smaller sides: \(14 + 16 = 30\). \(30\) is not greater than \(30\), so not a triangle.

Step4: Check \(\{14,27,40\}\)

Largest side \(c = 40\). Sum of smaller sides: \(14+27 = 41\). Since \(41>40\), check other inequalities (though largest side sum is key, but verify others too: \(14 + 40>27\) (54>27, true), \(27 + 40>14\) (67>14, true)).

Step5: Check \(\{11,24,36\}\)

Largest side \(c = 36\). Sum of smaller sides: \(11 + 24 = 35\). \(35<36\), so not a triangle.

Step6: Check \(\{14,16,30\}\) (again, but wait, \(14 + 16 = 30\), not greater, so no). Wait, earlier for \(\{14,27,40\}\): \(14+27 = 41>40\), \(14 + 40>27\), \(27 + 40>14\). Wait, but wait, the options: let's re - check. Wait, the user's options: maybe I made a mistake. Wait, no, let's re - evaluate. Wait, the set \(\{14,27,40\}\): \(14+27 = 41>40\), \(14 + 40=54>27\), \(27 + 40 = 67>14\). Now check the other options again:

• \(\{14,25,39\}\): \(14 + 25=39\), not greater, so invalid.
• \(\{14,16,30\}\): \(14 + 16 = 30\), not greater, invalid.
• \(\{11,24,36\}\): \(11+24 = 35<36\), invalid.
• \(\{14,27,40\}\): \(14 + 27=41>40\), so this satisfies the triangle inequality.

Answer:

\(\{14,27,40\}\)