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The problem asks for the equation of a line perpendicular to the line \( y = 4 \) that passes through the point \( (5, 2) \).

### Step-by-step solution:

1. **Understand the given line**: 
   The equation \( y = 4 \) represents a **horizontal line** where the slope is \( 0 \) (since it’s a constant \( y \)-value). The equation can be written as:
   \[
   y = 0x + 4
   \]
   The slope \( m_1 = 0 \).

2. **Perpendicular line**: 
   A line perpendicular to a horizontal line will be **vertical**, meaning its slope is undefined. A vertical line has the form:
   \[
   x = c
   \]
   where \( c \) is a constant.

3. **Find the equation of the perpendicular line**: 
   The line needs to pass through the point \( (5, 2) \), so the equation of the line is:
   \[
   x = 5
   \]

Thus, the equation of the line perpendicular to \( y = 4 \) and passing through \( (5, 2) \) is:
\[
x = 5
\]

Would you like any further details or have any questions?

Here are 5 related questions to further expand this concept:
1. What is the equation of a line perpendicular to \( y = -2 \) passing through the point \( (3, 7) \)?
2. How would you find the equation of a line parallel to \( y = 4 \) through a given point?
3. What is the general form of a vertical line's equation?
4. If a line is perpendicular to a vertical line, what is its slope?
5. How can you find the distance between two parallel lines?

**Tip**: For perpendicular lines, the product of their slopes is always \( -1 \), unless one of the lines is vertical or horizontal.

Find the equation of the line perpendicular to the line y = 4 containing the point (5, 2).

This problem explains how to find the equation of a line perpendicular to y = 4 passing through the point (5, 2). Explore the steps involved in understanding perpendicular lines, vertical line equations, and basic coordinate geometry concepts. Ideal for students in Grades 7-9.

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