Distance Between any Two Parallel Lines

Introduction

We can easily calculate the distance between any two parallel lines by the distance of a point from a line. Besides, it is equivalent to the length of the vertical distance from any point on one of the lines to another line. In this article, we discuss about the distance between the two lines’ derivation and formulas. Moreover, there are many benefits of this process to calculating the distance between two lines.

Key Takeaways:

Distance between two lines, Formulas, derivations

Distance Between Point and Line Derivation

We can express a line with a equation by Ax + By + C = 0. Consider a point P in the Cartesian plane having the coordinates (x1,y1). Besides, there are many things to calculate the distance from the point to the line. Moreover, in the Cartesian system, we can calculate the length of the perpendicular between the point and line.

The distance between the point P and the line L can be calculated by figuring out the length of the perpendicular as the figure given below.

Distance Between any Two Parallel Lines

The coordinate points for different points are as follows:

Point P (x1, y1), Point N (x2, y2), Point R (x3,y3)

The line L makes intercepts on both the x-axis and y – axis at the points N and M respectively. The coordinates of these points are M(0,−C/B)M(0,−C/B) and N (−C/A,0)N (−C/A,0).

The area of Δ MPN can be given as:

Area of Δ MPN = 1/2 × Base × Height

⇒Area of Δ MPN⇒Area of Δ MPN = 1/2 × PQ × MN

⇒PQ⇒PQ = 2 × Area of Δ MPN/MN ………………………(i)

In terms of Co-ordinate Geometry, the area of the triangle is given as:

Area of Δ MPN = 1/2[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]1/2[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]

Distance Between Two Parallel Lines

Furthermore, the distance between two parallel lines is equal to the perpendicular distance between the two lines. Thus, we know that the slopes of two parallel lines are the same. Therefore, the equation of two parallel lines can be given as:

y = mx + c1 and y = mx + c2

Besides, Point A is the intersection point of the second line on the x-axis.

The perpendicular distance would be the required distance between two lines

The distance between the point A and the line y = mx + c2 can be given by using the formula:

Shortest Distance Between Two Parallel Lines

Basically, the shortest distance between the two parallel lines can be determined using the length of the perpendicular segment between the lines. Besides, it does not matter which perpendicular line you are choosing, as long as two points are on the line. Thus, we can now easily calculate the distance between two parallel lines and the distance between a point and a line.

Things to Remember:

  • We can determine the distance between two lines through perpendicular between the point and line
  • The perpendicular line can be used for the identification of the length of the distance between two lines

 

Question: How can we calculate the distance between two parallel lines?

Ans: We can calculate the distance between two lines using a perpendicular between the point and line. It is the best method in the Cartesian system.

Question: What is the Role of a perpendicular line in calculating the distance between two parallel lines?  

Answer: The distance between point A and the line y = mx + c2 can be given by using the formula:

 

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