Write an equation of the line through the pointers (a) parallel to the given line (b)perpendicular to the given line Point( 2, 5) line x=4 

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 \( y = \text{constant} \).

Learn how to find equations of lines parallel and perpendicular to the line x = 4 passing through the point (2, 5). This problem explores geometric concepts involving vertical and horizontal lines in the coordinate plane. Suitable for Grades 9-12 students and anyone interested in geometry and line equations.

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