Let's find the equations of the lines that pass through the point $(2, 5)$ and are either:

1. Parallel to the given line $x = 4$.
2. Perpendicular to the given line $x = 4$.

Given Line

The given line is $x = 4$. This is a vertical line where the x-coordinate is always 4.

(a) Parallel Line

• A line parallel to $x = 4$ must also be a vertical line.
• Since the line needs to pass through the point $(2, 5)$, the equation of the parallel line will be $x = 2$.

(b) Perpendicular Line

• A line perpendicular to a vertical line is a horizontal line.
• A horizontal line has an equation of the form $y = \text{constant}$.
• Since the line needs to pass through the point $(2, 5)$, the equation of the perpendicular line will be $y = 5$.

Final Answer

• Equation of the line parallel to $x = 4$: $x = 2$
• Equation of the line perpendicular to $x = 4$: $y = 5$

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How do you write the equation of a vertical line?
2. What is the slope of a vertical line and a horizontal line?
3. How do you determine the equation of a line given a point and a slope?
4. How can you find the distance between two parallel lines?
5. How do you graph vertical and horizontal lines on the coordinate plane?

Tip: When dealing with vertical and horizontal lines, always remember that vertical lines have the equation $x = \text{constant}$, and horizontal lines have the equation $y = \text{constant}$.