Given \( T(8,-2), U(-8,2), V(-9,5) \), and \( W(x, 4) \). Find \( x \) such that \( \overline{T U} \| \overline{V W} \). Answer Attempt 1 out of 2

To find \( x \) such that \( \overline{T U} \| \overline{V W} \), we first need to determine the slopes of the lines \( \overline{TU} \) and \( \overline{VW} \). The slope of the line segment \( \overline{TU} \) between points \( T(8, -2) \) and \( U(-8, 2) \) is given by: \[ \text{slope of } \overline{TU} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-2)}{-8 - 8} = \frac{2 + 2}{-16} = \frac{4}{-16} = -\frac{1}{4} \] Next, we will find the slope of the line segment \( \overline{VW} \) between points \( V(-9, 5) \) and \( W(x, 4) \): \[ \text{slope of } \overline{VW} = \frac{4 - 5}{x - (-9)} = \frac{-1}{x + 9} \] Since \( \overline{TU} \) and \( \overline{VW} \) are parallel, their slopes must be equal: \[ -\frac{1}{4} = \frac{-1}{x + 9} \] To solve for \( x \), we cross-multiply: \[ -1 \cdot (x + 9) = -4 \cdot 1 \] This simplifies to: \[ -x - 9 = -4 \] Now, adding 9 to both sides gives us: \[ -x = 5 \quad \Rightarrow \quad x = -5 \] Thus, the value of \( x \) such that \( \overline{TU} \| \overline{VW} \) is \( \boxed{-5} \).