Question

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}
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Explanation:

Step1: Recall Triangle Inequality

For three lengths \(a\), \(b\), \(c\) to form a triangle, the sum of any two sides must be greater than the third side. i.e., \(a + b>c\), \(a + c>b\), \(b + c>a\).

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

Check \(14 + 25\) vs \(39\): \(14+25 = 39\), but \(39\) is not greater than \(39\) (needs to be \(>\)). So invalid.

Step3: Check \(\{14,27,40\}\)

Check \(14 + 27\) vs \(40\): \(14 + 27=41>40\); \(14 + 40 = 54>27\); \(27+40 = 67>14\). All inequalities hold.

Step4: Check \(\{14,16,30\}\)

Check \(14 + 16\) vs \(30\): \(14+16 = 30\), not greater. Invalid.

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

Check \(11 + 24\) vs \(36\): \(11+24 = 35<36\). Invalid.

Answer:

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