What is the Distance of a Point from a Line?

S6-SA1-0170

Grade Level:

Class 10

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

Definition

What is it?

The distance of a point from a line is the shortest possible distance between that point and any point on the line. This shortest distance is always measured along a perpendicular line segment connecting the point to the given line.

Simple Example

Quick Example

Imagine you are standing at a specific spot (the point) on a cricket field, and there's a straight boundary rope (the line) far away. The shortest way to run from where you are to the boundary rope is to run straight towards it, making a 90-degree angle with the rope. That straight path is the distance of your point from the line.

Worked Example

Step-by-Step

Let's find the distance of the point P(2, 3) from the line 3x + 4y - 5 = 0.

Step 1: Identify the coordinates of the point (x1, y1) and the coefficients of the line (A, B, C).
Here, (x1, y1) = (2, 3).
The line equation is Ax + By + C = 0, so A = 3, B = 4, C = -5.

---Step 2: Use the formula for the distance of a point from a line: d = |Ax1 + By1 + C| / sqrt(A^2 + B^2).

---Step 3: Substitute the values into the formula.
d = |(3)(2) + (4)(3) + (-5)| / sqrt((3)^2 + (4)^2).

---Step 4: Calculate the numerator.
Numerator = |6 + 12 - 5| = |18 - 5| = |13| = 13.

---Step 5: Calculate the denominator.
Denominator = sqrt(9 + 16) = sqrt(25) = 5.

---Step 6: Divide the numerator by the denominator to get the distance.
d = 13 / 5 = 2.6 units.

Answer: The distance of the point P(2, 3) from the line 3x + 4y - 5 = 0 is 2.6 units.

Why It Matters

This concept is crucial in fields like AI/ML for optimizing routes and positioning sensors, in Physics for understanding force fields, and in Engineering for designing structures and robots. Engineers use it to ensure parts fit correctly, and even in medicine, it helps analyze distances in medical imaging.

Common Mistakes

MISTAKE: Forgetting the absolute value in the numerator, leading to a negative distance. | CORRECTION: Distance is always a positive value. Always use the absolute value function (||) for the numerator to ensure the result is positive.

MISTAKE: Incorrectly identifying A, B, or C from the line equation, especially if it's not in the Ax + By + C = 0 form. | CORRECTION: Always rewrite the line equation into the standard form Ax + By + C = 0 before picking out the A, B, and C values.

MISTAKE: Making calculation errors in squaring or taking the square root in the denominator. | CORRECTION: Double-check all arithmetic, especially squaring the A and B values and finding the square root of their sum.

Practice Questions

Try It Yourself

QUESTION: Find the distance of the point (1, -2) from the line 4x - 3y + 7 = 0. | ANSWER: 3.8 units

QUESTION: Calculate the distance of the origin (0, 0) from the line x + y - 10 = 0. | ANSWER: 5 * sqrt(2) or approximately 7.07 units

QUESTION: A mobile tower is located at point T(5, 7). A straight highway passes through the points A(1, 2) and B(9, 6). Find the shortest distance from the mobile tower to the highway. (Hint: First find the equation of the line passing through A and B). | ANSWER: Approximately 2.83 units

MCQ

Quick Quiz

What is the formula for the distance of a point (x1, y1) from a line Ax + By + C = 0?

|Ax1 + By1 + C| / (A^2 + B^2)

(Ax1 + By1 + C) / sqrt(A^2 + B^2)

|Ax1 + By1 + C| / sqrt(A^2 + B^2)

sqrt(Ax1 + By1 + C) / (A + B)

The Correct Answer Is:

Option C is the correct formula. It correctly includes the absolute value in the numerator to ensure distance is positive and the square root of the sum of squares of A and B in the denominator.

Real World Connection

In the Real World

In urban planning, city engineers use this concept to calculate the shortest distance from a new building (point) to existing utility lines (line) to plan connections efficiently. Similarly, delivery apps like Zomato or Swiggy might use similar principles to estimate the shortest path from a delivery person's current location (point) to a straight road (line) to reach a customer faster.

Key Vocabulary

Key Terms

PERPENDICULAR: Forming a right angle (90 degrees) | COORDINATES: A set of numbers that show the exact position of a point on a graph | ORIGIN: The point (0,0) on a coordinate plane | ABSOLUTE VALUE: The non-negative value of a number, ignoring its sign (e.g., |-5| = 5)

What's Next

What to Learn Next

Great job understanding this! Next, you can explore 'Distance Between Two Parallel Lines.' This builds on what you've learned here, as it often involves finding the distance of a point on one line from the other line.