To find the equation of the line that is perpendicular to a given line and passes through a specific point, follow these steps:
1. Identify the Equation of the Given Line:
- The given lines are [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex]. These are vertical lines because their equations are in the form [tex]\( x = \text{constant} \)[/tex].
2. Determine the Slope of the Perpendicular Line:
- A line that is perpendicular to a vertical line must be a horizontal line. This is because vertical lines have undefined slopes, and the slopes of perpendicular lines are negative reciprocals of each other.
3. Understand the Characteristics of Horizontal Lines:
- Horizontal lines have equations of the form [tex]\( y = \text{constant} \)[/tex].
4. Identify the Given Point:
- The given point is [tex]\( (2, 6) \)[/tex]. This point must lie on the perpendicular line.
5. Find the Equation of the Perpendicular Line:
- Since the line must be horizontal and pass through the point [tex]\( (2, 6) \)[/tex], the y-coordinate of all points on this line must be 6. Therefore, the equation of the line is [tex]\( y = 6 \)[/tex].
So, the equation of the line that is perpendicular to the given vertical line [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex] and passes through the point [tex]\( (2, 6) \)[/tex] is
[tex]\[
y = 6
\][/tex]
