What are perpendicular lines?

Perpendicular lines are lines that intersect at a right angle (90°).

Think of:

• Corners of a square or rectangle

• Crossroads

• The edges of a piece of paper

Key property of perpendicular lines

The most important property is about slopes:

If two lines are perpendicular, the slope of one line is the negative reciprocal of the other.

Negative reciprocal explanation:

1. Take the slope of the first line: m₁

2. Flip it (reciprocal) → 1/m₁

3. Change the sign → negative

Example 1: Finding the perpendicular slope

Line:

• Slope of original line: m₁ = 2

• Negative reciprocal:

  • Flip: 1/2

  • Change sign: −1/2

✅ Slope of perpendicular line: m₂ = −1/2

Example 2: Writing a perpendicular line through a point

Given:
Line: y = −3x + 1
Point: (2, 4)

Step 1: Find perpendicular slope

• Original slope = −3

• Negative reciprocal = 1/3

Step 2: Use point-slope form

Step 3: Simplify (optional)

✅ This line is perpendicular to the original.

Special cases

1. Horizontal and vertical lines

   • Horizontal line: y = constant

   • Vertical line: x = constant

   • They are always perpendicular

2. Slope zero

   • Horizontal slope = 0 → perpendicular slope = undefined (vertical line)

Visual summary

Common beginner mistakes

• ❌ Forgetting to flip the slope before changing the sign
  ✅ Always do reciprocal first, then make negative

• ❌ Confusing perpendicular with parallel
  ✅ Parallel → same slope
  ✅ Perpendicular → negative reciprocal

• ❌ Forgetting special horizontal/vertical rules
  ✅ Horizontal ↔ vertical

Why perpendicular lines matter

• Geometry: angles, shapes, squares, rectangles

• Graphing: linear systems

• Real-world: construction, roads, design