Question

question
which of the following sets of numbers could represent the three sides of a triangle?
answer
{12, 19, 32} {8, 14, 24}
{5, 15, 21} {14, 24, 36}

Explanation:

Step1: Recall Triangle Inequality

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

Step2: Check \(\{12, 19, 32\}\)

Largest side \(32\). Sum of smaller: \(12 + 19 = 31\). Since \(31<32\), fails.

Step3: Check \(\{8, 14, 24\}\)

Largest side \(24\). Sum of smaller: \(8 + 14 = 22\). Since \(22<24\), fails.

Step4: Check \(\{5, 15, 21\}\)

Largest side \(21\). Sum of smaller: \(5 + 15 = 20\). Since \(20<21\), fails.

Step5: Check \(\{14, 24, 36\}\)

Largest side \(36\). Sum of smaller: \(14 + 24 = 38\). Since \(38>36\), and also check other sums: \(14 + 36>24\) (\(50>24\)) and \(24 + 36>14\) (\(60>14\)). So this set works.

Answer:

\(\{14, 24, 36\}\)