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) <x< Incorrect Apply the Triangle Inequality Theorem when given the lengths of two sides of a triangle. Using the Triangle Inequality Theorem, determine the range of lengths of the third side of the triangle.

Answer 1

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given two sides of lengths 34 and 51, we can denote the third side as \(x\). The Triangle Inequality Theorem gives us the following three inequalities:

1. \(34 + 51 > x\)
2. \(34 + x > 51\)
3. \(51 + x > 34\)

Now, let's solve each inequality:

1. \(34 + 51 > x\) \[ 85 > x \quad \text{or} \quad x < 85 \]

2. \(34 + x > 51\) \[ x > 51 - 34 \quad \Rightarrow \quad x > 17 \]

3. \(51 + x > 34\) \[ x > 34 - 51 \quad \Rightarrow \quad x > -17 \quad \text{(This inequality is always true since \(x\) is a length)} \]

Combining the relevant inequalities, we focus on the first two:

\[ 17 < x < 85 \]

Thus, the complete inequality for the third side \(x\) is:

\[ 17 < x < 85 \]