Distance Between Two Parallel Lines Calculator
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Distance Between Two Parallel Lines Calculator

Calculate Distance Between Two Parallel Lines Tools

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Second line:







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1. Distance Between Two Parallel Lines

When you have two parallel lines represented in the form of linear equations (Ax + By + C = 0), calculating the distance between them is essential. The distance is essentially the length of the perpendicular line segment connecting the two parallel lines.

Formula:

To find this distance, you can use the formula:

(d = |C1 - C2| / √(A2 + B2))

Where:

  • C1 and C2 are the constants in the linear equations representing the lines.
  • A and B are the coefficients of the x and y terms in the equations, respectively.

This formula essentially measures the perpendicular distance between the two lines. By taking the absolute difference between the constants C1 and C2, and dividing it by the square root of the sum of squares of the coefficients A and B, you get the distance between the two parallel lines.

2. Distance Between Two Parallel Lines

Understanding the Concept:

When dealing with parallel lines in a Euclidean space, understanding the distance between them is crucial. This distance is essentially the length of the perpendicular line segment connecting the two parallel lines.

Formula:

Let's denote the distance between two parallel lines as (d). To calculate this distance, you can use the following formula:

(d = |Ax + By + C| / √(A2 + B2))

Where:

  • Ax + By + C = 0 represents the equation of the first parallel line.
  • (x1, y1) denotes a point on the second parallel line.

Equivalent Formulas:

Point-Point Form:

(d = |Ax1 + By1 + C| / √(A2 + B2) = |Ax2 + By2 + C| / √(A2 + B2))

Where (x1, y1) and (x2, y2) are two points on the second parallel line.

Example:

Let's take two parallel lines with the following equations:

  1. 2x + 3y - 5 = 0
  2. 2x + 3y + 7 = 0

We'll calculate the distance between them using the formula mentioned above.

Calculation:

Given:

  • A = 2
  • B = 3
  • C1 = -5 (for the first line)
  • C2 = 7 (for the second line)

We'll take a point (x1, y1) on the second line, say (1, -2).

Substituting these values into the formula:

(d = |2(1) + 3(-2) + 7| / √(22 + 32))

(d = |2 - 6 + 7| / √(4 + 9))

(d = |3| / √13)

(d = 3 / √13)

So, the distance between the given parallel lines is 3 / √13 units.