What is the Distance Between Two Parallel Lines?

S6-SA1-0250

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The distance between two parallel lines is the shortest perpendicular distance between any point on one line and the other line. Since parallel lines never meet, this distance remains constant everywhere along their length.

Simple Example

Quick Example

Imagine two railway tracks running side-by-side. The distance between the two rails is always the same, no matter where you measure it along the track. This constant distance is the 'distance between two parallel lines'.

Worked Example

Step-by-Step

Let's find the distance between the parallel lines: 3x + 4y - 10 = 0 and 3x + 4y + 5 = 0.

Step 1: Identify the coefficients A, B, C1, and C2 from the general form Ax + By + C = 0.
For the first line: A = 3, B = 4, C1 = -10.
For the second line: A = 3, B = 4, C2 = 5.
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Step 2: Use the formula for the distance between two parallel lines: d = |C1 - C2| / sqrt(A^2 + B^2).
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Step 3: Substitute the values into the formula.
d = |-10 - 5| / sqrt(3^2 + 4^2)
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Step 4: Simplify the numerator and the denominator.
d = |-15| / sqrt(9 + 16)
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Step 5: Continue simplifying.
d = 15 / sqrt(25)
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Step 6: Calculate the square root.
d = 15 / 5
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Step 7: Perform the division.
d = 3

Answer: The distance between the two parallel lines is 3 units.

Why It Matters

Understanding the distance between parallel lines is crucial in fields like Engineering for designing stable structures, and in Physics for calculating forces. Even in Space Technology, it helps in planning satellite orbits to avoid collisions. Engineers and scientists use this concept daily.

Common Mistakes

MISTAKE: Using the formula only when coefficients of x and y are different | CORRECTION: Ensure the coefficients of x and y are identical in both equations before applying the formula. If they are not, multiply one equation by a suitable constant to make them identical.

MISTAKE: Forgetting the absolute value in the numerator | CORRECTION: The distance must always be a positive value. Remember to use the absolute value |C1 - C2| to ensure the result is positive.

MISTAKE: Incorrectly calculating the square root of (A^2 + B^2) | CORRECTION: Carefully calculate A^2 and B^2 first, then add them, and finally find the square root of the sum.

Practice Questions

Try It Yourself

QUESTION: Find the distance between the lines 2x + 3y - 6 = 0 and 2x + 3y + 4 = 0. | ANSWER: 10 / sqrt(13) units

QUESTION: What is the distance between the lines x + y = 7 and x + y = 3? | ANSWER: 4 / sqrt(2) units or 2*sqrt(2) units

QUESTION: Calculate the distance between the lines 6x + 8y - 15 = 0 and 3x + 4y + 10 = 0. (Hint: Make the coefficients of x and y same first) | ANSWER: 35 / 10 units or 3.5 units

MCQ

Quick Quiz

Which of the following formulas is used to find the distance between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0?

|C1 + C2| / sqrt(A^2 + B^2)

|C1 - C2| / sqrt(A^2 - B^2)

|C1 - C2| / sqrt(A^2 + B^2)

sqrt(A^2 + B^2) / |C1 - C2|

The Correct Answer Is:

The correct formula for the distance between two parallel lines is |C1 - C2| / sqrt(A^2 + B^2). Options A, B, and D have incorrect numerators or denominators.

Real World Connection

In the Real World

In urban planning, architects and civil engineers use this concept to ensure roads, bridges, and building foundations are parallel and maintain a safe, consistent distance. For example, when designing flyovers in cities like Bengaluru or Mumbai, engineers calculate these distances precisely to ensure smooth traffic flow and structural integrity.

Key Vocabulary

Key Terms

PARALLEL LINES: Lines that never intersect and maintain a constant distance between them | PERPENDICULAR DISTANCE: The shortest distance measured at a 90-degree angle | COEFFICIENTS: The numerical values multiplying the variables (like A and B in Ax + By + C = 0) | CONSTANT TERM: The term in an equation that does not contain any variables (like C in Ax + By + C = 0)

What's Next

What to Learn Next

Great job understanding parallel lines! Next, you should explore the 'Distance of a Point from a Line'. This will build on your current knowledge and help you solve more complex geometry problems, opening doors to understanding shapes and spaces better.