Question

which of the following can be the lengths of the sides of a triangle? 4, 9, 13 5, 6, 7 4, 4, 8 5\frac{1}{2}, 5\frac{1}{2}, 11

Explanation:

Step1: Check the triangle inequality for \(4,9,13\)

The triangle inequality states that for any triangle with side lengths \(a,b,c\), \(a + b>c\), \(a + c>b\) and \(b + c>a\).
For \(a = 4\), \(b=9\), \(c = 13\), \(4+9=13\), but \(a + b\) should be \(>\) \(c\). So, \(4,9,13\) does not form a triangle.

Step2: Check the triangle inequality for \(5,6,7\)

For \(a = 5\), \(b = 6\), \(c=7\)
\(5+6=11>7\)
\(5 + 7=12>6\)
\(6+7=13>5\)

Step3: Check the triangle inequality for \(4,4,8\)

For \(a = 4\), \(b = 4\), \(c = 8\), \(4+4=8\), but \(a + b\) should be \(>\) \(c\). So, \(4,4,8\) does not form a triangle.

Step4: Check the triangle inequality for \(5\frac{1}{2},5\frac{1}{2},11\)

\(5\frac{1}{2}+5\frac{1}{2}=11\), but \(a + b\) should be \(>\) \(c\). So, \(5\frac{1}{2},5\frac{1}{2},11\) does not form a triangle.

Answer:

\(5,6,7\)