Perpendicular distance between two equations? (parallel)

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SUMMARY

The discussion focuses on calculating the perpendicular distance between two parallel lines represented by the equations y=2x-1 and y=2x-8/3. The vertical distance between the y-intercepts of these lines is determined to be 5/3. To find the perpendicular distance, participants suggest deriving a third line that is perpendicular to both given lines, identifying the intersection points, and measuring the distance between these points. This method ensures that the calculated distance remains consistent regardless of the chosen point for the perpendicular line.

PREREQUISITES
  • Understanding of linear equations and their graphs
  • Knowledge of slopes and perpendicular lines in coordinate geometry
  • Familiarity with distance formulas in a Cartesian plane
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study the concept of slopes and how to determine perpendicular slopes
  • Learn how to derive equations of lines from given points and slopes
  • Explore the distance formula between two points in a coordinate system
  • Practice problems involving parallel and perpendicular lines in geometry
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Students studying geometry, mathematics educators, and anyone looking to understand the principles of calculating distances between parallel lines.

Dramacon
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Homework Statement


Find the perpendicular distance between y=2x-1 and y=2x - 8/3


2. The attempt at a solution
The first equation hits the y-axis at -1 and the other at -8/3, which means the vertical (y axis) distance between them is 5/3, but to use trig, I need another piece of information...this is where I am stuck... please help? :)
 
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Have you tried making a new line y_p perpendicular to both? The segment between the two y lines gives you your perpendicular distance.
 
Perpendicular distance between two equations? This depends upon where they are written on the page.

Now, to be serious ...

What is the slope of any line that's perpendicular to these two lines?
 
If SammyS hadn't beaten me to it, I would have said the same- "equations" are not geometric objects. You mean the distance between the two parallel lines that are the graphs, in a given coordinate system, of those two equations.

In any case, as both aeroplane and SammyS have said, find the equation of a line perpendicular to both lines. Find the two points where this third line crosses the two given lines and find the distance between those two points. Since that distance will be the same for any such perpendicular, you can choose whatever point you want for the line to go through.
 

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