To find the equation of the set of points that are equidistant from the parallel lines \( x = 1 \) and \( x = 7 \), we need to understand the concept of distance from a point to a line and how to find a midpoint between two parallel lines. ### Step-by-Step Explanation 1. **Understanding the Lines**: - The lines given are \( x = 1 \) and \( x = 7 \). These are vertical lines in the Cartesian coordinate system. - The line \( x = 1 \) is located at 1 unit on the x-axis, and the line \( x = 7 \) is located at 7 units on the x-axis. 2. **Finding the Midpoint**: - To find the set of points that are equidistant from both lines, we need to find the midpoint between these two lines. - The midpoint \( M \) between two points \( a \) and \( b \) on the x-axis can be calculated using the formula: \[ M = \frac{a + b}{2} \] - Here, \( a = 1 \) and \( b = 7 \): \[ M = \frac{1 + 7}{2} = \frac{8}{2} = 4 \] - This means that the line \( x = 4 \) is equidistant from both \( x = 1 \) and \( x = 7 \). 3. **Understanding the Set of Points**: - The equation \( x = 4 \) represents a vertical line where every point on this line has an x-coordinate of 4. - Since the lines \( x = 1 \) and \( x = 7 \) are vertical, the distance from any point on the line \( x = 4 \) to both lines is the same. 4. **Conclusion**: - Therefore, the equation of the set of points that are equidistant from the lines \( x = 1 \) and \( x = 7 \) is \( x = 4 \). ### Why Other Options Are Incorrect - **Option A: \( y = 3 \)**: - This represents a horizontal line where all points have a y-coordinate of 3. It does not relate to the distance from the vertical lines \( x = 1 \) and \( x = 7 \). - **Option B: \( x = 3 \)**: - This line is located to the left of the midpoint (4) and is not equidistant from both lines. The distance from \( x = 3 \) to \( x = 1 \) is 2 units, while the distance to \( x = 7 \) is 4 units. - **Option D: \( y = 4 \)**: - Similar to option A, this represents a horizontal line where all points have a y-coordinate of 4. It does not provide any information about the distance from the vertical lines. ### Summary of Key Points - The midpoint between two parallel lines can be found using the average of their positions. - The equation \( x = 4 \) represents the set of points equidistant from the lines \( x = 1 \) and \( x = 7 \). - Vertical lines are defined by their x-coordinates, while horizontal lines are defined by their y-coordinates, which are not relevant in this context. - Always check the distances from proposed solutions to ensure they meet the criteria of being equidistant. Thus, the correct answer is **C. \( x = 4 \)**.