Welcome to Coordinate Geometry: Perpendicular Lines!
Hi everyone! In the world of coordinate geometry, lines can interact in many ways. They can be parallel (running side-by-side) or they can intersect. But when two lines meet in a very special way—forming a perfect right angle—we call them perpendicular lines.
This chapter is vital because understanding perpendicularity allows you to solve complex problems involving shapes, distances, and constructions on the coordinate plane. Don't worry if this seems tricky at first; we will break down the one critical rule you need to know using simple steps and fun tricks!
1. Quick Review: What is a Gradient?
Before diving into perpendicular lines, let's quickly remember the star of coordinate geometry: the gradient.
The equation of a straight line is written as:
\(y = mx + c\)
- \(m\) is the gradient (slope or steepness).
- \(c\) is the \(y\)-intercept (where the line crosses the \(y\)-axis).
Prerequisite: Parallel Lines
Remember that if two lines are parallel, they have the same gradient.
If Line 1 has gradient \(m_1\) and Line 2 has gradient \(m_2\), then for parallel lines:
\(m_1 = m_2\)
Example: A line parallel to \(y = 5x - 3\) must also have a gradient of 5.
Key Takeaway (Review)
The gradient \(m\) dictates the slope. Parallel lines share the same \(m\).
2. The Golden Rule of Perpendicular Lines
Two lines are perpendicular if and only if the product of their gradients is -1.
The Formula
If Line 1 has gradient \(m_1\) and Line 2 has gradient \(m_2\), then for them to be perpendicular:
\(m_1 \times m_2 = -1\)
The "Negative Reciprocal" Trick
While the formula works, it's often easier and faster to find the perpendicular gradient (\(m_2\)) by using the concept of the negative reciprocal.
To find the negative reciprocal of a number, follow two steps:
- Flip it: Find the reciprocal (turn the fraction upside down).
- Change the sign: If it was positive, make it negative. If it was negative, make it positive.
Analogy: Think of perpendicular lines like turning a corner. You don't just walk the same slope (parallel); you flip your direction (reciprocal) and reverse the steepness (negative).
Examples of Negative Reciprocals
- If \(m_1 = 3\), then \(m_2 = -\frac{1}{3}\)
- If \(m_1 = -\frac{2}{5}\), then \(m_2 = \frac{5}{2}\)
- If \(m_1 = -1\), then \(m_2 = 1\)
- If \(m_1 = \frac{4}{7}\), then \(m_2 = -\frac{7}{4}\)
Quick Review: The perpendicular gradient is the negative reciprocal of the original gradient.
3. Step-by-Step: Finding the Perpendicular Gradient
Step 1: Ensure the Equation is in \(y = mx + c\) form
You must rearrange the given line equation to isolate \(y\) so you can clearly identify the gradient \(m\).
Example: Find the gradient of a line perpendicular to \(2y = 3x + 1\).
We must first find the gradient of the given line (\(m_1\)):
\(2y = 3x + 1\)
(Divide everything by 2)
\(y = \frac{3}{2}x + \frac{1}{2}\)
So, the original gradient is \(m_1 = \frac{3}{2}\).
Step 2: Calculate the Negative Reciprocal
Now, we find \(m_2\) (the perpendicular gradient) by flipping \(\frac{3}{2}\) and changing the sign:
\(m_2 = -\frac{2}{3}\)
Did you know? This relationship ensures the lines meet at exactly \(90^\circ\). You can test this by multiplying the gradients: \(\frac{3}{2} \times (-\frac{2}{3}) = -1\).
Common Mistake to Avoid:
If you are given \(ax + by = c\), you cannot just pick out a number. You must rearrange to \(y = mx + c\) first!
4. Finding the Equation of a Perpendicular Line
The goal is usually not just to find the gradient, but to find the full equation of a new perpendicular line that passes through a specific point.
Example: Finding the Equation
Find the equation of the line that is perpendicular to \(y = 2x + 5\) and passes through the point \((4, 1)\).
Step 1: Find the Perpendicular Gradient (\(m_2\))
The original line is \(y = 2x + 5\), so \(m_1 = 2\).
The perpendicular gradient (\(m_2\)) is the negative reciprocal of 2 (or \(\frac{2}{1}\)).
\(m_2 = -\frac{1}{2}\)
Step 2: Use the Gradient and the Point to Find the \(y\)-intercept (\(c\))
We use the general form \(y = mx + c\).
- We know \(m = -\frac{1}{2}\).
- We know the line passes through \((x, y) = (4, 1)\).
Substitute these values into the equation:
\(1 = \left(-\frac{1}{2}\right)(4) + c\)
\(1 = -2 + c\)
\(c = 1 + 2\)
\(c = 3\)
Step 3: Write the Final Equation
Combine your perpendicular gradient (\(m_2 = -\frac{1}{2}\)) and your new \(y\)-intercept (\(c = 3\)).
The equation of the perpendicular line is: \(y = -\frac{1}{2}x + 3\)
Key Takeaway (Equation)
Finding the perpendicular equation requires two pieces of information: the negative reciprocal gradient and a point that helps you locate the line (find \(c\)).
5. Advanced Application: The Perpendicular Bisector
This concept combines both perpendicular lines and the midpoint formula (from syllabus E4.3). This is a common Extended content question.
A perpendicular bisector of a line segment is a line that:
- Is perpendicular (at \(90^\circ\)) to the segment.
- Bisects the segment (cuts it exactly in half, passing through the midpoint).
Prerequisite: Midpoint Formula
Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \(M\) is:
\(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
Step-by-Step: Finding the Perpendicular Bisector
Example: Find the equation of the perpendicular bisector of the line segment joining \(A(-3, 8)\) and \(B(9, -2)\).
Step 1: Find the Midpoint (The "Bisector" part)
\(x\)-coordinate of Midpoint: \(\frac{-3 + 9}{2} = \frac{6}{2} = 3\)
\(y\)-coordinate of Midpoint: \(\frac{8 + (-2)}{2} = \frac{6}{2} = 3\)
The midpoint is \(M = (3, 3)\). This is the point your new line must pass through.
Step 2: Find the Gradient of the Original Line Segment (\(m_{AB}\))
The gradient formula is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
\(m_{AB} = \frac{-2 - 8}{9 - (-3)} = \frac{-10}{12} = -\frac{5}{6}\)
Step 3: Find the Perpendicular Gradient (\(m_2\))
The perpendicular gradient is the negative reciprocal of \(-\frac{5}{6}\).
\(m_2 = \frac{6}{5}\)
Step 4: Find the Equation of the Perpendicular Bisector
Use the gradient \(m_2 = \frac{6}{5}\) and the midpoint \((3, 3)\) in the form \(y = mx + c\):
\(3 = \left(\frac{6}{5}\right)(3) + c\)
\(3 = \frac{18}{5} + c\)
\(c = 3 - \frac{18}{5}\)
\(c = \frac{15}{5} - \frac{18}{5} = -\frac{3}{5}\)
The equation of the perpendicular bisector is: \(y = \frac{6}{5}x - \frac{3}{5}\)
Quick Review (Perpendicular Bisector)
A perpendicular bisector problem has two parts: finding the midpoint (where it cuts) and finding the negative reciprocal gradient (how it cuts).