Parallel and Perpendicular Lines

Parallel and Perpendicular Lines#

Slope and Line Equations Revision#

Theory#

Before studying parallel and perpendicular lines, recall:

  • Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

  • Slope-intercept form: \(y = mx + c\)

  • Lines with the same slope have the same steepness and direction

Application#

Two lines with slopes \(m_1 = 2\) and \(m_2 = 2\) will never intersect - they are parallel.

Parallel and Perpendicular Lines#

Theory#

Parallel Lines#

Two lines are parallel if and only if:

  • They have the same slope: \(m_1 = m_2\)

  • They never intersect

  • They maintain constant distance apart

For lines in general form:

  • \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) are parallel if \(\frac{a_1}{b_1} = \frac{a_2}{b_2}\)

Perpendicular Lines#

Two lines are perpendicular if and only if:

  • The product of their slopes equals -1: \(m_1 \times m_2 = -1\)

  • They intersect at a 90° angle

  • If one slope is \(m\), the perpendicular slope is \(-\frac{1}{m}\)

Special cases:

  • A horizontal line (slope = 0) is perpendicular to a vertical line (undefined slope)

  • For lines in general form: \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) are perpendicular if \(a_1a_2 + b_1b_2 = 0\)

Interactive Visualization: Parallel and Perpendicular Lines Explorer#

Application#

Examples#

Example 1: Finding a parallel line

Find the equation of the line parallel to \(y = 3x - 2\) passing through the point \((4, 1)\).

Solution:

  • Parallel lines have the same slope, so \(m = 3\)

  • Using point-slope form: \(y - 1 = 3(x - 4)\)

  • Simplifying: \(y - 1 = 3x - 12\)

  • Therefore: \(y = 3x - 11\)

Example 2: Finding a perpendicular line

Find the equation of the line perpendicular to \(2x - 5y + 10 = 0\) passing through \((3, -1)\).

Solution:

  • First, find the slope of the given line:

    • Rearrange to slope-intercept form: \(5y = 2x + 10\)

    • \(y = \frac{2}{5}x + 2\)

    • Slope \(m_1 = \frac{2}{5}\)

  • Perpendicular slope: \(m_2 = -\frac{1}{m_1} = -\frac{5}{2}\)

  • Using point-slope form: \(y - (-1) = -\frac{5}{2}(x - 3)\)

  • \(y + 1 = -\frac{5}{2}x + \frac{15}{2}\)

  • \(y = -\frac{5}{2}x + \frac{13}{2}\)

Example 3: Determining line relationships

Determine whether the following pairs of lines are parallel, perpendicular, or neither: a) \(y = 2x + 3\) and \(y = 2x - 5\) b) \(3x + 4y = 12\) and \(4x - 3y = 8\) c) \(y = \frac{1}{3}x + 2\) and \(y = -3x + 1\)

Solution:

a) Slopes: \(m_1 = 2\), \(m_2 = 2\) Since \(m_1 = m_2\), the lines are parallel.

b) Convert to slope-intercept form:

  • Line 1: \(4y = -3x + 12\), so \(y = -\frac{3}{4}x + 3\), thus \(m_1 = -\frac{3}{4}\)

  • Line 2: \(3y = 4x - 8\), so \(y = \frac{4}{3}x - \frac{8}{3}\), thus \(m_2 = \frac{4}{3}\)

  • Product: \(m_1 \times m_2 = -\frac{3}{4} \times \frac{4}{3} = -1\) The lines are perpendicular.

c) Slopes: \(m_1 = \frac{1}{3}\), \(m_2 = -3\)

  • Product: \(\frac{1}{3} \times (-3) = -1\) The lines are perpendicular.

Multiple Choice Questions#

Sector Specific Questions: Parallel and Perpendicular Applications#

Key Takeaways#

Important

Essential Parallel and Perpendicular Concepts:

  1. Parallel Lines:

    • Same slope: \(m_1 = m_2\)

    • Never intersect

    • General form: \(\frac{a_1}{b_1} = \frac{a_2}{b_2}\)

  2. Perpendicular Lines:

    • Product of slopes = -1: \(m_1 \times m_2 = -1\)

    • Intersect at 90°

    • If one slope is \(m\), the other is \(-\frac{1}{m}\)

    • General form: \(a_1a_2 + b_1b_2 = 0\)

  3. Special Cases:

    • Horizontal (m = 0) ⊥ Vertical (undefined m)

    • Vertical lines: parallel if same x-intercept

  4. Applications: Construction, design, physics, navigation

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