Question

charla has six segments with which to make two triangles. the segments lengths are 2 in., 3 in., 4 in., 5 in., 6 in., and 7 in. which are possible side lengths of her two triangles?
2 in., 4 in., 6 in. and 3 in., 5 in., 7 in.
2 in., 5 in., 6 in. and 3 in., 4 in., 7 in.
2 in., 3 in., 4 in. and 5 in., 6 in., 7 in.
2 in., 3 in., 6 in. and 4 in., 5 in., 7 in.

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Step2: Check option 1

For the set \(2\) in., \(4\) in., \(6\) in.: \(2 + 4=6\), which does not satisfy the triangle - inequality theorem. So, this option is incorrect.

Step3: Check option 2

For the set \(2\) in., \(5\) in., \(6\) in.: \(2+5 > 6\), \(2 + 6>5\), \(5 + 6>2\). For the set \(3\) in., \(4\) in., \(7\) in.: \(3+4 = 7\), which does not satisfy the triangle - inequality theorem. So, this option is incorrect.

Step4: Check option 3

For the set \(2\) in., \(3\) in., \(4\) in.: \(2+3>4\), \(2 + 4>3\), \(3 + 4>2\). For the set \(5\) in., \(6\) in., \(7\) in.: \(5+6>7\), \(5 + 7>6\), \(6 + 7>5\). This option satisfies the triangle - inequality theorem.

Step5: Check option 4

For the set \(2\) in., \(3\) in., \(6\) in.: \(2+3<6\), which does not satisfy the triangle - inequality theorem. So, this option is incorrect.

Answer:

C. \(2\) in., \(3\) in., \(4\) in. and \(5\) in., \(6\) in., \(7\) in.