The slope of a line in the standard (x,y) coordinate plane is 4. What is the slope of a line perpendicular to that line ?

Options

A. 4
B. `1/4`
C. `-1/4`
D. `-1`

Answer

To find the slope of a line that is perpendicular to a given line with a slope of 4, we can follow these steps:

### Step 1: Understand the relationship between the slopes of perpendicular lines.
The product of the slopes of two perpendicular lines is equal to -1. If we denote the slope of the first line as \( m_1 \) and the slope of the second line as \( m_2 \), we can express this relationship mathematically as:
\[ m_1 \times m_2 = -1 \]

### Step 2: Identify the slope of the given line.
In this case, the slope of the given line (let's call it line \( L_1 \)) is:
\[ m_1 = 4 \]

### Step 3: Set up the equation using the relationship from Step 1.
We can substitute the value of \( m_1 \) into the equation:
\[ 4 \times m_2 = -1 \]

### Step 4: Solve for the slope of the perpendicular line.
To find \( m_2 \), we can rearrange the equation:
\[ m_2 = \frac{-1}{4} \]

### Step 5: State the result.
Thus, the slope of the line that is perpendicular to the line with a slope of 4 is:
\[ m_2 = -\frac{1}{4} \]

### Final Answer:
The slope of the line perpendicular to the line with slope 4 is \(-\frac{1}{4}\).

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