Question

which of the following sets of numbers could represent the three sides of a triangle? answer {15,19,34} {9,13,19} {8,20,30} {9,24,34} submit answer

Explanation:

Step1: Recall Triangle Inequality Theorem

The sum of any two sides of a triangle must be greater than the third side. For a set of three numbers \(a\), \(b\), \(c\) (where \(c\) is the largest), we need \(a + b>c\), \(a + c>b\), and \(b + c>a\). Since \(a + c>b\) and \(b + c>a\) are always true if \(a + b>c\) (because \(c\) is the largest, \(a + c>b\) and \(b + c>a\) will hold as long as \(a + b>c\)). So we just need to check \(a + b>c\) for each set, where \(c\) is the largest number.

Step2: Check set \(\{15, 19, 34\}\)

Largest number is \(34\). Check \(15 + 19\). \(15+19 = 34\). But the sum should be greater than \(34\), not equal. So this set does not form a triangle.

Step3: Check set \(\{9, 13, 19\}\)

Largest number is \(19\). Check \(9 + 13\). \(9 + 13=22\). Since \(22>19\), and also \(9 + 19>13\) (\(28>13\)) and \(13 + 19>9\) (\(32>9\)), this set can form a triangle. Let's check the other sets to be sure.

Step4: Check set \(\{8, 20, 30\}\)

Largest number is \(30\). Check \(8 + 20\). \(8+20 = 28\). Since \(28<30\), this set does not form a triangle.

Step5: Check set \(\{9, 24, 34\}\)

Largest number is \(34\). Check \(9 + 24\). \(9+24 = 33\). Since \(33<34\), this set does not form a triangle.

Answer:

\(\{9, 13, 19\}\)