What is the Equation of a Perpendicular Line? | Class 10

S6-SA1-0304

What is the Equation of a Line Passing Through a Point and Perpendicular to another Line?

Grade Level:

Class 10

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

Definition

What is it?

The equation of a line passing through a specific point and perpendicular to another given line is found by first determining the slope of the given line. Then, we use the negative reciprocal of that slope to find the slope of our new perpendicular line. Finally, we use the point-slope form with the given point and the new perpendicular slope to write the equation.

Simple Example

Quick Example

Imagine you're drawing a straight road (Line A) on a map. Now, you need to draw a new shortcut road (Line B) that starts from your friend's house (a specific point) and crosses the first road at a perfect 90-degree angle. Finding the equation of Line B is like figuring out the exact path of this shortcut road.

Worked Example

Step-by-Step

Find the equation of a line passing through the point (2, 3) and perpendicular to the line 2x + 3y = 5.

STEP 1: Find the slope of the given line.
The given line is 2x + 3y = 5. Rearrange it into y = mx + c form.
3y = -2x + 5
y = (-2/3)x + 5/3
The slope of the given line (m1) is -2/3.

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STEP 2: Find the slope of the perpendicular line.
If two lines are perpendicular, the product of their slopes is -1. So, m1 * m2 = -1.
(-2/3) * m2 = -1
m2 = -1 / (-2/3)
m2 = 3/2.
The slope of the perpendicular line is 3/2.

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STEP 3: Use the point-slope form to find the equation of the new line.
The point-slope form is y - y1 = m(x - x1). We have the point (x1, y1) = (2, 3) and the slope m = 3/2.
y - 3 = (3/2)(x - 2)

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STEP 4: Simplify the equation.
Multiply both sides by 2 to remove the fraction:
2(y - 3) = 3(x - 2)
2y - 6 = 3x - 6

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STEP 5: Rearrange into standard form (Ax + By + C = 0).
3x - 2y - 6 + 6 = 0
3x - 2y = 0

ANSWER: The equation of the line is 3x - 2y = 0.

Why It Matters

This concept is super important in fields like engineering and robotics, where machines need to move along specific paths and make precise turns. For example, self-driving cars use this to navigate roads and avoid obstacles by calculating perpendicular distances. It's also vital in designing buildings (architecture) and even in computer graphics for creating realistic 3D models.

Common Mistakes

MISTAKE: Using the same slope for the perpendicular line as the given line. | CORRECTION: Remember that perpendicular lines have slopes that are negative reciprocals of each other (m1 * m2 = -1). If m1 is 2, m2 is -1/2.

MISTAKE: Incorrectly rearranging the given line equation to find its slope (e.g., mixing up signs). | CORRECTION: Always convert the equation to the y = mx + c form carefully. Pay close attention to positive and negative signs when moving terms across the equals sign.

MISTAKE: Substituting the given point (x1, y1) into the 'c' part of y = mx + c instead of using the point-slope form or solving for 'c' correctly. | CORRECTION: After finding the perpendicular slope (m), use the point-slope form (y - y1 = m(x - x1)) directly or substitute (x1, y1) and 'm' into y = mx + c to find 'c'.

Practice Questions

Try It Yourself

QUESTION: Find the equation of the line passing through (1, 5) and perpendicular to y = 2x + 7. | ANSWER: x + 2y - 11 = 0

QUESTION: A line passes through the point (-3, 4) and is perpendicular to the line joining points (1, 2) and (5, 0). Find its equation. | ANSWER: 2x - y + 10 = 0

QUESTION: The vertices of a triangle are A(1, 2), B(4, 2), and C(3, 5). Find the equation of the altitude from vertex A to side BC. (Hint: An altitude is perpendicular to the opposite side). | ANSWER: x - 3y + 5 = 0

MCQ

Quick Quiz

What is the slope of a line perpendicular to the line x - 4y + 8 = 0?

-4

2026-01-04T00:00:00.000Z

-1/4

The Correct Answer Is:

First, find the slope of the given line: x - 4y + 8 = 0 => 4y = x + 8 => y = (1/4)x + 2. The slope is 1/4. The slope of a perpendicular line is the negative reciprocal, which is -4. So, option B is correct.

Real World Connection

In the Real World

Think about how delivery apps like Zomato or Swiggy optimize routes. They use concepts of lines and perpendicularity to calculate the shortest distance from a delivery person to a customer, or to find the most efficient turn. Similarly, civil engineers use this to design precise road intersections and railway tracks, ensuring safety and smooth traffic flow.

Key Vocabulary

Key Terms

SLOPE: A measure of the steepness of a line. | PERPENDICULAR LINES: Two lines that intersect at a 90-degree angle. | NEGATIVE RECIPROCAL: If a slope is 'm', its negative reciprocal is '-1/m'. | POINT-SLOPE FORM: A way to write a linear equation: y - y1 = m(x - x1).

What's Next

What to Learn Next

Great job understanding perpendicular lines! Next, you can explore finding the equation of a line parallel to another line. This builds on similar concepts of slope but involves a slightly different relationship between the slopes of the two lines.