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)%0D%0A <x<%0D%0A<%0D%0A𝑥%0D%0A<

Answer 1

To determine the range of possible values for the third side \( x \) of a triangle when the other two sides measure 34 and 51, we can use 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.

This gives us the following inequalities:

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

Now, let's simplify each inequality:

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

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

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

Combining these results, we have: \[ 17 < x < 85 \]

Therefore, the complete inequality that indicates the possible values of the third side \( x \) is: \[ 17 < x < 85 \]