Lesson: Determining Distance Between Two Parallel Lines on the Coordinate Plane

Key Concepts

Shortest Distance: The shortest distance between two parallel lines is the length of a perpendicular segment.
Perpendicular Segments: Infinite perpendicular segments can exist between parallel lines. We aim to find a perpendicular segment with integer coordinates for its endpoints.
Perpendicular Slopes: Slopes of perpendicular lines are negative reciprocals of each other.
- Example: If one line has a slope of ( -\frac{2}{3} ), the perpendicular line would have a slope of ( \frac{3}{2} ).
- Exception: Horizontal (slope = 0) and vertical lines (slope = undefined) are always perpendicular.

Finding a Convenient Point
- Choose a convenient point on either line with integer coordinates.
- Example given: point with coordinates (2, -3).
Determine the Slope of the Line
- Use the formula ( m = \frac{y_2 - y_1}{x_2 - x_1} ) or select two points to draw a triangle showing rise over run.
- Example: Change in y = 2, change in x = 4, slope = ( \frac{1}{2} ).
Find the Perpendicular Segment
- The perpendicular segment will have a slope that is the negative reciprocal of the line’s slope.
- For slope ( \frac{1}{2} ), negative reciprocal is (-2) or (-\frac{2}{1}).
- Sketch a line with this slope passing through the chosen point, using the slope to find endpoints.
- Example: Moving down 2, right 1 or up 2, left 1 to find intersection points with integer coordinates.
Calculate the Distance
- Use the distance formula: ( \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
- Example: From (2, -3) to (0, 1), ( \text{distance} = \sqrt{20} = 2\sqrt{5} \approx 4.472 ).

If the intersection point is not easily determined with integer coordinates, solve the system of equations for the lines to find it.

Understanding the relationship between slopes and how to use the distance formula is crucial in calculating the shortest distance between parallel lines on a coordinate plane.