Question

14 which set of integers could represent the lengths of the sides of an isosceles triangle? 1) {1,1,3} 2) {2,2,5} 3) {3,3,6} 4) {4,4,7}

Explanation:

Step1: Check triangle inequality for {1,1,3}

For a triangle with side lengths \(a = 1\), \(b = 1\), \(c = 3\), check \(a + b>c\).
\(1+1 = 2<3\). So, this is not a triangle.

Step2: Check triangle inequality for {2,2,5}

For \(a = 2\), \(b = 2\), \(c = 5\), check \(a + b>c\).
\(2 + 2=4<5\). So, this is not a triangle.

Step3: Check triangle inequality for {3,3,6}

For \(a = 3\), \(b = 3\), \(c = 6\), check \(a + b>c\).
\(3+3 = 6\). Since \(a + b=c\), this is not a triangle.

Step4: Check triangle inequality for {4,4,7}

For \(a = 4\), \(b = 4\), \(c = 7\)

• Check \(a + b>c\): \(4 + 4=8>7\)
• Check \(a + c>b\): \(4+7 = 11>4\)
• Check \(b + c>a\): \(4 + 7=11>4\)

Also, since two sides (\(a = 4\) and \(b = 4\)) are equal, it is an isosceles triangle.

Answer:

1. \(\{4,4,7\}\)