Question

question
which of the following sets of numbers could represent the three sides of a triangle?
answer
{13,22,34}
{8,22,30}
{7,12,20}
{6,9,15}
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Explanation:

Step1: Recall Triangle Inequality

For three lengths \(a\), \(b\), \(c\) (where \(c\) is the largest), the triangle inequality states \(a + b>c\).

Step2: Check \(\{13, 22, 34\}\)

Largest side is \(34\). Sum of other two: \(13 + 22=35\). Since \(35>34\), check other inequalities ( \(13 + 34>22\) and \(22 + 34>13\) which are true, but main check is smallest two sum > largest). Wait, no, wait: Wait, \(13+22 = 35>34\), but let's check other sets too.

Step3: Check \(\{7, 12, 20\}\)

Largest side \(20\). Sum of \(7 + 12 = 19\). Since \(19<20\), this fails.

Step4: Check \(\{8, 22, 30\}\)

Largest side \(30\). Sum of \(8 + 22=30\). But \(30\) is not greater than \(30\) (needs to be \(>\)), so fails.

Step5: Check \(\{6, 9, 15\}\)

Largest side \(15\). Sum of \(6 + 9 = 15\). Again, not greater, fails. Wait, wait, earlier step2: Wait, \(13 + 22 = 35>34\), so \(13\), \(22\), \(34\) satisfy \(a + b>c\) (since \(13+22>34\), \(13 + 34>22\), \(22 + 34>13\)). Wait, but let's re - check:

Wait, for \(\{13,22,34\}\):

• \(13 + 22=35>34\)
• \(13 + 34 = 47>22\)
• \(22 + 34=56>13\)

For \(\{7,12,20\}\): \(7 + 12 = 19<20\) → fails.

For \(\{8,22,30\}\): \(8 + 22 = 30\) (not greater) → fails.

For \(\{6,9,15\}\): \(6 + 9=15\) (not greater) → fails.

So the set \(\{13,22,34\}\) satisfies the triangle inequality.

Answer:

\(\{13, 22, 34\}\)