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

S6-SA1-0265

What is the Equation of a Line Perpendicular to a Given Line?

Grade Level:

Class 10

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

Definition

What is it?

The equation of a line perpendicular to a given line describes a line that crosses the given line at a perfect 90-degree angle. The key idea is that the product of their slopes will always be -1.

Simple Example

Quick Example

Imagine two streets crossing each other in your neighbourhood. If one street runs perfectly east-west and the other perfectly north-south, they are perpendicular. If you know the 'slope' or 'steepness' of one street, you can easily find the slope of the other street that crosses it at a right angle.

Worked Example

Step-by-Step

Let's find the equation of a line perpendicular to the line 2x + 3y = 6, passing through the point (4, 1).
---Step 1: Convert the given equation to slope-intercept form (y = mx + c) to find its slope.
2x + 3y = 6
3y = -2x + 6
y = (-2/3)x + 2
---Step 2: Identify the slope (m1) of the given line.
The slope m1 = -2/3.
---Step 3: Calculate the slope (m2) of the perpendicular line. Remember, m1 * m2 = -1.
(-2/3) * m2 = -1
m2 = -1 / (-2/3)
m2 = 3/2
---Step 4: Use the point-slope form (y - y1 = m(x - x1)) with the perpendicular slope (m2) and the given point (4, 1).
y - 1 = (3/2)(x - 4)
---Step 5: Simplify the equation to slope-intercept form (y = mx + c) or standard form (Ax + By = C).
y - 1 = (3/2)x - (3/2)*4
y - 1 = (3/2)x - 6
y = (3/2)x - 6 + 1
y = (3/2)x - 5
---Answer: The equation of the perpendicular line is y = (3/2)x - 5.

Why It Matters

Understanding perpendicular lines is crucial for designing sturdy buildings and bridges in Engineering. In Physics, it helps calculate forces acting at right angles, like pushing a cart up a ramp. Doctors use similar concepts in Medicine for precise measurements in imaging.

Common Mistakes

MISTAKE: Students often forget to take the negative reciprocal of the slope, just using the reciprocal. | CORRECTION: The perpendicular slope is the negative reciprocal. If original slope is 'm', perpendicular slope is '-1/m'.

MISTAKE: Incorrectly converting the given line equation into slope-intercept form (y = mx + c), leading to a wrong original slope. | CORRECTION: Always isolate 'y' carefully, making sure to divide all terms by the coefficient of 'y'.

MISTAKE: Mixing up the coordinates (x1, y1) when substituting into the point-slope form. | CORRECTION: Double-check that you are putting the x-coordinate in the x-place and the y-coordinate in the y-place.

Practice Questions

Try It Yourself

QUESTION: What is the slope of a line perpendicular to a line with slope 5? | ANSWER: -1/5

QUESTION: Find the equation of a line perpendicular to y = 4x + 7 and passing through the point (8, 3). | ANSWER: y = (-1/4)x + 5

QUESTION: A line passes through points A(2, 5) and B(6, 1). Find the equation of a line perpendicular to line AB and passing through the midpoint of AB. | ANSWER: y = x - 2

MCQ

Quick Quiz

Which of the following lines is perpendicular to the line y = -3x + 2?

y = 3x + 5

y = (1/3)x - 1

y = -3x + 7

y = -(1/3)x + 4

The Correct Answer Is:

The slope of the given line is -3. The slope of a perpendicular line must be the negative reciprocal, which is -1/(-3) = 1/3. Option B has a slope of 1/3.

Real World Connection

In the Real World

When civil engineers design flyovers or bridges in cities like Delhi or Mumbai, they often need to ensure that supporting beams or connecting roads are perpendicular to each other for stability. Similarly, in robotics, robot arms need to move in precisely perpendicular directions for tasks like picking up items or assembling parts.

Key Vocabulary

Key Terms

PERPENDICULAR: Two lines or objects meeting at a 90-degree angle | SLOPE: The steepness of a line, often represented by 'm' | RECIPROCAL: Flipping a fraction (e.g., reciprocal of 2/3 is 3/2) | NEGATIVE RECIPROCAL: The reciprocal with an opposite sign | SLOPE-INTERCEPT FORM: An equation of a line written as y = mx + c

What's Next

What to Learn Next

Great job learning about perpendicular lines! Next, you can explore 'Parallel Lines' where lines never meet and have the same slope. This will help you understand how lines behave differently based on their slopes.