5.3 Parallel and Perpendicular Lines
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Learning Objectives
- Identify when two lines are parallel or perpendicular using gradients
- Find the equations of lines that are parallel or perpendicular to a given line
- Apply these principles to solve geometric and contextual problems
Key Concepts
Parallel Lines
- Two lines are **parallel** if they have the same gradient (slope).
- If Line 1 has gradient *m*, and Line 2 has the same gradient *m*, they are parallel.
- Example:
- Line A: y = 2x + 3
- Line B: y = 2x - 4
→ Both lines have gradient 2, so they are parallel.
Perpendicular Lines
- Two lines are **perpendicular** if the product of their gradients is -1.
- That is, if Line 1 has gradient *m*, then Line 2 must have gradient **-1/m**.
- Example:
- Line A: y = 3x + 2 → Gradient = 3
- Line B: y = (-1/3)x + 5 → Gradient = -1/3
→ 3 × (-1/3) = -1 → Lines are perpendicular.
Finding Equations of Parallel/Perpendicular Lines
- To find the equation of a line **parallel** to y = mx + c and passing through (x₁, y₁):
- Use the same gradient m and apply point-slope form:
→ y - y₁ = m(x - x₁)
- To find the equation of a line **perpendicular** to y = mx + c and passing through (x₁, y₁):
- Use the gradient -1/m instead.
→ y - y₁ = (-1/m)(x - x₁)
- Rearranged into y = mx + c form if needed.
Geometrical Applications
- These techniques are essential in coordinate geometry problems like:
- Proving lines are parallel or perpendicular
- Finding altitudes or normals in triangle geometry
- Constructing geometric shapes such as rectangles or squares
- Often used alongside midpoints and distances.
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