Line Properties and Equations

Overview

This lecture covers the properties and equations of parallel and perpendicular lines, how to find their gradients, and how to apply these concepts to solve various line equation problems.

Parallel Lines

• Parallel lines are indicated with arrows or notation (e.g., AB ∥ CD).
• The gradients (slopes) of parallel lines are always equal.
• To prove lines are parallel, calculate both gradients and show they are the same.
• Example: If lines through (6,4)-(7,11) and (0,0)-(1,7) both have gradient 7, they are parallel.

Perpendicular Lines

• Perpendicular lines meet at a 90° angle and are denoted AB ⟂ CD.
• The product of the gradients of two perpendicular lines is -1.
• The gradient of a perpendicular line is the negative reciprocal of the original gradient.
• Example transformations:
  • Gradient -2 → perpendicular gradient ½,
  • Gradient ¼ → perpendicular gradient -4,
  • Gradient 1 → perpendicular gradient -1,
  • Gradient -2/3 → perpendicular gradient 3/2.

Finding Specific Line Equations

• To find the equation of a parallel line, use the same gradient and the given point.
• Convert the given equation to slope-intercept (y=mx+c) form to find the gradient.
• Use point-gradient form: y - y₁ = m(x - x₁).
• For perpendicular lines, find the negative reciprocal of the gradient and apply point-gradient form.

Perpendicular Bisector

• A perpendicular bisector cuts a line segment at its midpoint at a right angle.
• First, find the gradient of the segment; then take the perpendicular gradient.
• Find the midpoint by averaging x and y values of the endpoints.
• Use the midpoint as the point in the equation with the perpendicular gradient.

Key Terms & Definitions

• Gradient (Slope) — Change in y divided by change in x (rise over run).
• Parallel Lines — Lines that never meet and have equal gradients.
• Perpendicular Lines — Lines that intersect at 90°; their gradients multiply to -1.
• Negative Reciprocal — Flip the fraction and change its sign.
• Midpoint — Average of the coordinates of two endpoints.
• Perpendicular Bisector — A line that divides a segment into two equal parts at 90°.

Action Items / Next Steps

• Complete Exercise 1i on page 59 of your textbook.