This section provides an exploration of perpendicular lines in coordinate geometry, emphasizing calculations and equations over textual explanations. We'll cover how to determine the gradient and equation of lines perpendicular to a given line, including the method for finding the perpendicular bisector of a line segment.
![Perpendicular lines](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F7be21180-b8f0-4fee-babc-623ceaeef397-file.png&w=1200&q=75)
Image courtesy of Tutor
Understanding Gradients and Perpendicularity
Gradient formula:
Perpendicular condition:
![Slopes of Perpendicular lines](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fc707610d-de5e-43d0-b47e-94dbf0ef977a-file.png&w=828&q=75)
Image courtesy of CueMath
Example 1: Gradient of a Perpendicular Line
Given a line with equation , determine the gradient of a line perpendicular to it.
- Original gradient:
- Perpendicular gradient:
Thus, the perpendicular gradient is .
Equation of Perpendicular Lines
Example 2: Finding Equation of Perpendicular Line
Determine the equation of a line perpendicular to that passes through .
- Given gradient:
- Perpendicular gradient:
- Point:
- Equation:
- Substituting values:
Thus, the equation is .
Graphical Representation:
![Graph of Perpendicular Lines](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F453a57f9-7a25-4cbe-97ae-5f4e6186bbbe-file.png&w=1080&q=75)
Finding the Perpendicular Bisector
Example 3: Perpendicular Bisector Between Points and
- Midpoint of :
- Gradient of :
- Perpendicular gradient:
- Using and , equation:
The perpendicular bisector of the segment is .
Graphical Representation:
![Perpendicular Bisector](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F7b8008f3-2757-45f5-80ca-6a78fdbea736-file.png&w=1080&q=75)