unit 8

Geometry of Straight Lines

Introduction

  • The study of straight lines involves concepts like gradient (slope), inclination, and equations of straight lines.

Gradient (Slope)

  • The gradient (slope) of a line is defined as the tangent of the inclination angle ( [ m = an(\theta) ] )

  • Common angles and their slopes:

    • 0°:

      • Gradient: [ m = \tan(0°) = 0 ]

    • 30°:

      • Gradient: [ m = \tan(30°) \approx 0.577 ]

    • 45°:

      • Gradient: [ m = \tan(45°) = 1 ]

    • 60°:

      • Gradient: [ m = \tan(60°) \approx 1.732 ]

    • 120°:

      • Gradient: [ m = \tan(120°) \approx -1.732 ]

    • 150°:

      • Gradient: [ m = \tan(150°) \approx -0.577 ]

    • 170°:

      • Gradient: [ m = \tan(170°) \approx -0.176 ]

Inclination of Lines

  • The inclination angle of a line can be calculated using the inverse tangent function: [ \theta = \tan^{-1}(m) ]

Finding the Inclination

  1. Calculate based on Gradient:

    • Example: For [ m = 0.577 ]

      • [ \theta = \tan^{-1}(0.577) \approx 30° ]

  2. Specific slopes:

    • [ m = \tan(45.5°) \approx 1.018 ]

Exercise 8.1

  • Find the gradient (slope) given inclination:

    1. 0°: m = 0

    2. 30°: m \approx 0.577

    3. 60°: m \approx 1.732

    4. 90°: m = ∞ (undefined)

Examples of Finding Line Gradient

  • Finding gradient from coordinates:

    • Coordinates A(2,6) B(5,8): [ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 6}{5 - 2} = \frac{2}{3} ]

    • Coordinates C(-2,4) D(1,-3): [ m = \frac{-3 - 4}{1 + 2} = \frac{-7}{3} ]

Parallel and Perpendicular Lines

  • Lines A and B:

    • Parallel lines have the same slope.

    • Perpendicular lines have slopes that are negative reciprocals: [ m_A * m_B = -1 ]

Proving Collinearity

  • Points A, B, C are collinear if the slopes between any two pairs are equal.

Example: Angle Between Two Lines

  1. Finding angle between two lines:

    • Use tangent formula: [ \theta = \tan^{-1} \left( \frac{m_2 - m_1}{1 + m_2 m_1} \right) ]

    • Calculate to find the angle between lines.

Summary

  • Understanding straight lines involves calculating their slopes, finding inclinations, and solving related exercises to establish relationships between given coordinates and their gradients.