To find the equation of a line that is perpendicular to $x = 1$ and passes through the point $(4, -5)$:

Step 1: Identify the characteristics of the line $x = 1$

• The line $x = 1$ is a vertical line.
• A line perpendicular to a vertical line must be a horizontal line.

Step 2: Write the equation of the horizontal line

• A horizontal line has an equation of the form $y = c$, where $c$ is a constant representing the $y$-coordinate of every point on the line.

Step 3: Use the point $(4, -5)$ to determine $c$

• Since the line must pass through $(4, -5)$, the $y$-coordinate of this point is $-5$.
• Thus, the equation of the line is: $y = -5$

Final Answer:

The equation of the line perpendicular to $x = 1$ and passing through $(4, -5)$ is: $y = -5$

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Would you like a graphical representation or further explanation?

Related Questions:

1. What is the slope of a line parallel to $x = 1$?
2. How do you find a perpendicular line to a given horizontal line?
3. Can you derive a perpendicular line equation in general form $Ax + By = C$?
4. How do vertical and horizontal lines differ geometrically?
5. How can you find the intersection point of a vertical and horizontal line?

Tip:

Remember that vertical lines always have undefined slopes, while horizontal lines always have a slope