Finding the gradient (slope) of a perpendicular line is a fundamental concept in coordinate geometry. Mastering this skill is crucial for solving various geometry problems and understanding linear relationships. This structured plan will guide you through the process, ensuring you grasp the concept thoroughly.
Understanding Gradients
Before diving into perpendicular lines, let's refresh our understanding of gradients. The gradient of a line represents its steepness or slope. It's calculated as the change in the y-coordinates divided by the change in the x-coordinates between any two points on the line. The formula is:
m = (y₂ - y₁) / (x₂ - x₁)
where:
- m represents the gradient
- (x₁, y₁) and (x₂, y₂) are two points on the line.
Example:
Let's say we have two points, (2, 4) and (6, 10). The gradient is:
m = (10 - 4) / (6 - 2) = 6 / 4 = 3/2 = 1.5
This means the line rises 1.5 units for every 1 unit it moves horizontally.
The Relationship Between Gradients of Perpendicular Lines
The key to finding the gradient of a perpendicular line lies in understanding their relationship. Two lines are perpendicular if they intersect at a right angle (90°). The gradients of perpendicular lines are negatively reciprocal to each other. This means:
m₁ * m₂ = -1
Where:
- m₁ is the gradient of the first line.
- m₂ is the gradient of the perpendicular line.
This relationship is crucial. If you know the gradient of one line, you can easily find the gradient of its perpendicular line by following these steps:
- Find the reciprocal: Flip the fraction (or change the decimal to a fraction and then flip).
- Change the sign: If the gradient is positive, make it negative; if it's negative, make it positive.
Example:
If the gradient of a line (m₁) is 2, the gradient of the perpendicular line (m₂) is:
- Reciprocal of 2 is 1/2
- Change the sign: -1/2
Therefore, the gradient of the perpendicular line is -1/2.
Finding the Gradient of a Perpendicular Line: A Step-by-Step Guide
Let's put this into practice with a detailed example.
Problem: Find the gradient of the line perpendicular to the line passing through points (1, 3) and (4, 9).
Solution:
-
Calculate the gradient of the original line:
m₁ = (9 - 3) / (4 - 1) = 6 / 3 = 2
-
Find the reciprocal:
The reciprocal of 2 is 1/2
-
Change the sign:
The gradient is positive, so we make it negative: -1/2
Therefore, the gradient of the line perpendicular to the line passing through (1,3) and (4,9) is -1/2.
Practice Problems
To solidify your understanding, try these practice problems:
- Find the gradient of the line perpendicular to a line with a gradient of -3.
- A line passes through points (-2, 1) and (4, 7). Find the gradient of the perpendicular line.
- If the gradient of a line is 0, what is the gradient of the perpendicular line? (Think carefully about this one!)
By working through these examples and practice problems, you'll develop a strong understanding of how to find the gradient of a perpendicular line. Remember the key relationship: m₁ * m₂ = -1. This formula is your best friend in tackling these types of problems.
