Find the perpendicular line to the line Y+9=0 (-5,8)

To find the equation of the line that is perpendicular to the given line \( Y + 9 = 0 \) and passes through the point \( (-5, 8) \), we will follow these steps:

### Step 1: Interpret the equation of the given line
The equation \( Y + 9 = 0 \) simplifies to:
\[
Y = -9
\]
This represents a horizontal line with a constant \( y \)-value of \( -9 \).

### Step 2: Determine the slope of the given line
Since the given line is horizontal, its slope (\( m_1 \)) is:
\[
m_1 = 0
\]

### Step 3: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Since the slope of the given line is \( 0 \), the slope of the perpendicular line (\( m_2 \)) is undefined. This means the perpendicular line is a vertical line.

### Step 4: Equation of the vertical line
A vertical line passing through the point \( (-5, 8) \) will have a constant \( x \)-value. Therefore, the equation of the perpendicular line is:
\[
x = -5
\]

### Final Answer:
The equation of the line perpendicular to \( Y + 9 = 0 \) that passes through \( (-5, 8) \) is:
\[
x = -5
\]

Do you want more details or have any questions?

Here are 5 related questions for practice:
1. What is the equation of a line perpendicular to \( y = 3 \) passing through \( (2, -4) \)?
2. Find the equation of a line parallel to \( y = -2x + 5 \) through \( (0, 0) \).
3. How do you calculate the distance from a point to a vertical line?
4. If a line has a slope of \( \frac{3}{4} \), what is the slope of a line perpendicular to it?
5. What is the equation of a vertical line passing through \( (7, -1) \)?

**Tip:** For vertical lines, the equation is always of the form \( x = \text{constant} \), while for horizontal lines, it's \( y = \text{constant} \).

Discover how to find the equation of a line perpendicular to the horizontal line Y + 9 = 0 that passes through the point (-5, 8). This solution covers key concepts in coordinate geometry and perpendicular lines, suitable for students in Grades 8-10.

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