# Distance Between Parallel Lines

### Distance Between Parallel Lines

#### Introduction:

• The distance between parallel lines refers to the shortest distance measured perpendicular from any point on one line to the other parallel line.

#### Formula for Distance between Parallel Lines:

• Given two parallel lines:
1. Line 1: $Ax+By+{C}_{1}=0$
2. Line 2: $Ax+By+{C}_{2}=0$
• The distance 'd' between these parallel lines is given by:
$\text{Distance}=\frac{\mathrm{\mid }{C}_{2}-{C}_{1}\mathrm{\mid }}{\sqrt{{A}^{2}+{B}^{2}}}$

#### Steps to Find Distance:

1. Calculate Numerator:
• Subtract the constants (intercepts) of the equations of the parallel lines: $\mathrm{\mid }{C}_{2}-{C}_{1}\mathrm{\mid }$.
2. Calculate Denominator:
• Use the coefficients of $x$ and $y$ in the line equations to find $\sqrt{{A}^{2}+{B}^{2}}$
• .
3. Divide Numerator by Denominator:
• Divide $\mathrm{\mid }{C}_{2}-{C}_{1}\mathrm{\mid }$ by $\sqrt{{A}^{2}+{B}^{2}}$
• to determine the distance.

#### Example:

Given two parallel lines:

1. Line 1: $3x+4y-5=0$

2. Line 2: $3x+4y+7=0$

3. Calculate Numerator:

• $\mathrm{\mid }{C}_{2}-{C}_{1}\mathrm{\mid }=\mathrm{\mid }7-\left(-5\right)\mathrm{\mid }=\mathrm{\mid }12\mathrm{\mid }$
4. Calculate Denominator:

• Use coefficients of $x$ and $y$: $\sqrt{{3}^{2}+{4}^{2}}=\sqrt{9+16}=\sqrt{25}=5$
1.
2. Find Distance:

• Distance $=\frac{\mathrm{\mid }12\mathrm{\mid }}{5}=\frac{12}{5}=2.4$ units

#### Importance:

• Determines the shortest distance between two parallel lines.
• Crucial in geometry, architecture, and engineering for accurate measurements and spatial planning.

#### Summary:

• The distance between parallel lines is computed using the difference in their constants divided by the square root of the sum of the squares of their coefficients.
• This measurement provides critical information for spatial analysis, optimization, and design in various fields.