Find the distance between two parallel lines.
Understand the Problem
The question is asking for the distance between two parallel lines, which involves using the formula for the distance between parallel lines in a coordinate plane.
Answer
The distance between parallel lines is $d = \frac{|b_2 - b_1|}{\sqrt{1 + m^2}}$.
Answer for screen readers
The distance between the two parallel lines is given by
$$ d = \frac{|b_2 - b_1|}{\sqrt{1 + m^2}} $$
Substituting specific values will yield the numerical distance.
Steps to Solve
- Identify the equations of the lines
The distance between two parallel lines can be calculated if we have their equations. For example, consider the lines given in the forms:
$$ y = m x + b_1 $$
and
$$ y = m x + b_2 $$
where $m$ is the slope and $b_1$, $b_2$ are the y-intercepts.
- Use the distance formula for parallel lines
The distance $d$ between the two parallel lines can be given by the formula:
$$ d = \frac{|b_2 - b_1|}{\sqrt{1 + m^2}} $$
This formula measures the vertical distance between the two lines.
- Plug in the values
If specific values for $b_1$ and $b_2$ are given, substitute them into the formula along with the slope $m$.
- Calculate the result
Perform the necessary arithmetic to find the value of $d$.
The distance between the two parallel lines is given by
$$ d = \frac{|b_2 - b_1|}{\sqrt{1 + m^2}} $$
Substituting specific values will yield the numerical distance.
More Information
For any two parallel lines, their distance remains constant regardless of how far you extend the lines. This distance can be visualized as the shortest path perpendicular to both lines.
Tips
- Mixing up $b_1$ and $b_2$: Ensure you correctly identify the y-intercepts for both lines.
- Forgetting to take the absolute value: Always use $|b_2 - b_1|$ to avoid negative distances.
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