To find the equations of the lines that are perpendicular to the given lines and pass through the point [tex]\((2, 6)\)[/tex], let's consider each given line individually.
1. Given Line: [tex]\( x = 2 \)[/tex]
- This is a vertical line passing through all points where [tex]\( x = 2 \)[/tex].
- A line perpendicular to a vertical line is horizontal.
- The horizontal line passing through the point [tex]\((2, 6)\)[/tex] must have the same [tex]\( y \)[/tex]-coordinate of 6.
- Therefore, the equation of the line is [tex]\( y = 6 \)[/tex].
2. Given Line: [tex]\( x = 6 \)[/tex]
- This is another vertical line passing through all points where [tex]\( x = 6 \)[/tex].
- Similar to the previous case, a line perpendicular to a vertical line is horizontal.
- The horizontal line passing through the point [tex]\((2, 6)\)[/tex] must have the same [tex]\( y \)[/tex]-coordinate of 6.
- Therefore, the equation of the line is [tex]\( y = 6 \)[/tex].
3. Given Line: [tex]\( y = 2 \)[/tex]
- This is a horizontal line passing through all points where [tex]\( y = 2 \)[/tex].
- A line perpendicular to a horizontal line is vertical.
- The vertical line passing through the point [tex]\((2, 6)\)[/tex] must have the same [tex]\( x \)[/tex]-coordinate of 2.
- Therefore, the equation of the line is [tex]\( x = 2 \)[/tex].
4. Given Line: [tex]\( y = 6 \)[/tex]
- This is another horizontal line passing through all points where [tex]\( y = 6 \)[/tex].
- Similar to the previous case, a line perpendicular to a horizontal line is vertical.
- The vertical line passing through the point [tex]\((2, 6)\)[/tex] must have the same [tex]\( x \)[/tex]-coordinate of 2.
- Therefore, the equation of the line is [tex]\( x = 2 \)[/tex].
In summary, the equations of the lines that are perpendicular to the given lines and pass through the point [tex]\((2, 6)\)[/tex] are:
- [tex]\( y = 6 \)[/tex] for the lines perpendicular to [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex].
- [tex]\( x = 2 \)[/tex] for the lines perpendicular to [tex]\( y = 2 \)[/tex] and [tex]\( y = 6 \)[/tex].
