Two sides of a triangle measure 34 and 51. Complete the inequality that indicates the possible values of the third side of the triangle using the Triangle Inequality Theorem.(1 point)

Question

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Answer 1

To find the possible values of the third side of a triangle, we can apply the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the lengths of the two known sides be \(a = 34\) and \(b = 51\), and let \(c\) represent the length of the third side. The Triangle Inequality Theorem gives us the following inequalities:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Substituting the known values into these inequalities:

1. \(34 + 51 > c\) → \(85 > c\) or \(c < 85\)
2. \(34 + c > 51\) → \(c > 51 - 34\) → \(c > 17\)
3. \(51 + c > 34\) → This inequality will always be true for positive values of \(c\), so it does not need further consideration.

Combining the inequalities from the first two results, we have:

\[17 < c < 85\]

Therefore, the completed inequality indicating the possible values of the third side \(c\) of the triangle is:

\[17 < c < 85\]