Welcome to Coordinate Geometry: Perpendicular Lines!

Hello! You’ve already mastered the basics of straight lines—finding gradients, lengths, and equations. This chapter builds on that foundation by introducing a really useful concept: lines that meet at a perfect right angle.

In the world of coordinate geometry, perpendicular lines have a special, predictable relationship between their gradients. Mastering this relationship is key to solving complex problems, like finding the centre of a circle or calculating the shortest distance between points.

Section 1: What does Perpendicular Mean?

You might remember this term from geometry (Section 5 of the syllabus).

Definition of Perpendicular Lines

Two lines are perpendicular if they intersect (cross) at a right angle (\(90^\circ\)).
Think of the corner of a square desk or the two intersecting axes (x-axis and y-axis) on your graph paper—these are perfect examples of perpendicular lines.

The Key Rule: Gradients of Perpendicular Lines

This is the most important concept in this entire section. When two lines are perpendicular, their gradients are linked by a simple, elegant rule.

Let \(m_1\) be the gradient of the first line, and \(m_2\) be the gradient of the second line.

The rule states that the product of their gradients must equal \(-1\):

$$m_1 \times m_2 = -1$$

If you know the gradient of one line (\(m_1\)), you can easily find the gradient of the line perpendicular to it (\(m_2\)).

How to Find the Perpendicular Gradient (\(m_2\))

The gradient of the perpendicular line is the negative reciprocal of the original gradient.

$$m_2 = -\frac{1}{m_1}$$

Memory Aid: The "Flip It and Change the Sign" Trick!

To get the negative reciprocal, you must do two things:

  1. Flip It (Reciprocal): Turn the fraction upside down.
  2. Change the Sign (Negative): If the original gradient was positive, the new one is negative (and vice versa).

Example: Applying the trick

  • If the original gradient \(m_1 = 3\) (which is \(\frac{3}{1}\)),
    Flip it: \(\frac{1}{3}\). Change the sign: \(m_2 = -\frac{1}{3}\).
  • If the original gradient \(m_1 = -\frac{2}{5}\),
    Flip it: \(\frac{5}{2}\). Change the sign: \(m_2 = \frac{5}{2}\).
  • If the original gradient \(m_1 = -1\),
    Flip it: \(\frac{1}{1}\) (still 1). Change the sign: \(m_2 = 1\).

Did you know? This rule only works because we use a Cartesian coordinate system where the axes are themselves perpendicular!

Quick Review: Parallel vs. Perpendicular

We often mix these up—be careful!

  • Parallel Lines: Have the SAME gradient. (\(m_1 = m_2\))
  • Perpendicular Lines: Have the NEGATIVE RECIPROCAL gradient. (\(m_1 \times m_2 = -1\))

Section 2: Finding the Equation of a Perpendicular Line

To find the equation of any straight line, you need two things: the gradient (\(m\)) and a point (\(x_1, y_1\)) that the line passes through.

The final equation must be given in the form \(y = mx + c\).

Step-by-Step Process

Let's find the equation of a line L2 that is perpendicular to the line \(y = 2x + 5\) and passes through the point \((4, -1)\).

Step 1: Find the Gradient of the Original Line (\(m_1\))

  • The original line is \(y = 2x + 5\).
  • The gradient \(m_1\) is the coefficient of \(x\).
    \(m_1 = 2\).

Step 2: Calculate the Perpendicular Gradient (\(m_2\))

  • Use the negative reciprocal rule (Flip it and Change the Sign).
  • \(m_1 = \frac{2}{1}\). Flip and change sign: \(m_2 = -\frac{1}{2}\).

Step 3: Substitute \(m_2\) and the given point into \(y = mx + c\)

  • We know \(m = -\frac{1}{2}\) and the line passes through \((4, -1)\).
  • Substitute \(x=4\), \(y=-1\), and \(m=-\frac{1}{2}\) into \(y = mx + c\):
    \(-1 = (-\frac{1}{2})(4) + c\)

Step 4: Solve for the y-intercept (\(c\))

  • \(-1 = -2 + c\)
  • \(c = -1 + 2\)
  • \(c = 1\)

Step 5: Write the Final Equation

  • Using \(m = -\frac{1}{2}\) and \(c = 1\), the equation for L2 is:
    $$y = -\frac{1}{2}x + 1$$

Key Takeaway: Finding a perpendicular equation always involves first finding the negative reciprocal gradient, and then using the given point to find the specific \(c\) value for that line.

⚠ Common Mistake to Avoid!

If the original equation is not in the form \(y = mx + c\), you must rearrange it first!
For example, if the line is \(3x + 2y = 6\):

1. Rearrange: \(2y = -3x + 6\)
2. Divide by 2: \(y = -\frac{3}{2}x + 3\)
3. The original gradient \(m_1\) is \(-\frac{3}{2}\).
4. The perpendicular gradient \(m_2\) is then \(\frac{2}{3}\). (Flip and change sign!)

Section 3: The Perpendicular Bisector (Extended Content)

This is the most common and comprehensive type of perpendicular line problem you will face. A perpendicular bisector is a line that cuts a line segment (a section of a line, like the line connecting point A to point B) exactly in half AND at a right angle.

To find the equation of a perpendicular bisector, you need to use two fundamental coordinate geometry concepts:

  1. The Midpoint Formula (to ensure it bisects the segment).
  2. The Negative Reciprocal Rule (to ensure it is perpendicular).

Prerequisite Formulas (Quick Review)

For a line segment joining point A \((x_1, y_1)\) and point B \((x_2, y_2)\):

1. Gradient \(m\) (E4.2): $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

2. Midpoint M (E4.3): $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Finding the Equation of a Perpendicular Bisector (Step-by-Step)

Let's find the perpendicular bisector of the line segment joining A \((-3, 8)\) and B \((9, -2)\). (Syllabus Example E4.6)

Step 1: Find the Midpoint (M) of AB

  • This is the specific point \((x, y)\) that the new line must pass through.
  • $$M = \left(\frac{-3 + 9}{2}, \frac{8 + (-2)}{2}\right)$$
  • $$M = \left(\frac{6}{2}, \frac{6}{2}\right) = (3, 3)$$

Step 2: Find the Gradient of the Original Segment (AB)

  • $$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 8}{9 - (-3)}$$
  • $$m_{AB} = \frac{-10}{12} = -\frac{5}{6}$$

Step 3: Calculate the Perpendicular Gradient (\(m_{\text{perp}}\))

  • Use the negative reciprocal rule on \(m_{AB} = -\frac{5}{6}\).
  • Flip it: \(\frac{6}{5}\). Change the sign: \(m_{\text{perp}} = \frac{6}{5}\).

Step 4: Find the Equation using \(y = mx + c\)

  • We use \(m = \frac{6}{5}\) and the point \(M(3, 3)\) from Step 1.
  • Substitute into \(y = mx + c\):
    $$3 = \left(\frac{6}{5}\right)(3) + c$$
  • $$3 = \frac{18}{5} + c$$
  • To solve for \(c\), convert 3 to fifths: \(3 = \frac{15}{5}\).
  • $$c = \frac{15}{5} - \frac{18}{5} = -\frac{3}{5}$$

Step 5: Write the Final Equation

  • $$y = \frac{6}{5}x - \frac{3}{5}$$
  • (Note: Sometimes the question asks for the answer in a different form, like \(ax + by = c\). If so, multiply by 5: \(5y = 6x - 3\), or \(6x - 5y = 3\).)

Key Takeaway for Perpendicular Bisectors: A bisector means "find the midpoint first!" The midpoint is the essential point needed to solve for \(c\).

Chapter Summary: Perpendicular Lines
  1. Perpendicular lines meet at \(90^\circ\).
  2. Their gradients are negative reciprocals: \(m_1 m_2 = -1\).
  3. To find the equation of a perpendicular line, find \(m_{\text{perp}}\), then use the given point to find \(c\) in \(y = mx + c\).
  4. To find a perpendicular bisector, you must calculate the midpoint first. This midpoint then serves as the essential point for calculating the equation.