Equations of Parallel and Perpendicular Lines

Equations of Parallel and Perpendicular Lines

Parallel Lines

Parallel lines are two lines in the same plane that never intersect. Understanding their properties is essential for writing equations of lines with specific relationships.

• Definition: Parallel lines are lines in the same plane that do not intersect.

• Slopes of Parallel Lines: Parallel lines have the same slope.

• Vertical Lines: If two lines are both vertical (with undefined slopes), they are parallel.

Key Properties:

• If two lines have the same slope, then they are parallel.

• Vertical lines (e.g., $x = a$) are parallel to each other.

Example:

• Given a line with equation $y = 2x + 3$, a parallel line passing through the point $(1, 4)$ will also have slope $m = 2$.

• Point-Slope Form: $y - 4 = 2(x - 1)$

• Slope-Intercept Form: $y = 2x + 2$

Perpendicular Lines

Perpendicular lines are lines that intersect to form a right angle (90 degrees). Their slopes have a special relationship.

• Definition: Perpendicular lines are lines that intersect at a right angle.

• Slopes of Perpendicular Lines: The slopes are negative reciprocals of each other.

• Horizontal and Vertical Lines: A horizontal line (slope $0$) is perpendicular to a vertical line (undefined slope).

Key Properties:

• If the product of the slopes of two lines is $-1$, the lines are perpendicular.

• If one line has slope $m$, a perpendicular line has slope $-\frac{1}{m}$ (provided $m \neq 0$).

Example:

• Given a line with equation $y = -\frac{1}{3}x + 5$, a perpendicular line passing through $(2, 1)$ will have slope $m = 3$.

• Point-Slope Form: $y - 1 = 3(x - 2)$

• Slope-Intercept Form: $y = 3x - 5$

Forms of Linear Equations

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

• General Form: $Ax + By + C = 0$

Writing Equations Given a Point and a Relationship

1. Identify the slope: For parallel lines, use the same slope as the given line. For perpendicular lines, use the negative reciprocal.

2. Use the given point: Substitute the point $(x_1, y_1)$ into the point-slope form.

3. Convert to desired form: Rearrange to slope-intercept or general form as needed.

Summary Table: Slopes of Parallel and Perpendicular Lines

Relationship	Slope of Line 1 ($m_1$)	Slope of Line 2 ($m_2$)	Condition
Parallel	$m_1$	$m_2$	$m_1 = m_2$
Perpendicular	$m_1$	$m_2$	$m_1 \times m_2 = -1$
Vertical & Horizontal	Undefined	0	Perpendicular

Practice Problems (from original notes):

• Write the equation of a line passing through a given point and parallel or perpendicular to a given line, in point-slope, slope-intercept, or general form.

• Examples include finding equations for lines passing through specified points and with specified relationships to other lines.

Additional info: The original notes referenced specific examples (problems 5–12) without providing the actual points or equations. The above examples and explanations illustrate the general process for such problems.