3.4 Equations of parallel and perpendicular lines

Parallel and perpendicular lines are key players in geometry. They help us understand relationships between lines and solve real-world problems. Knowing how to find slopes and write equations for these lines is crucial.

We'll learn to identify parallel and perpendicular lines using slopes and equations. This knowledge is super useful for tackling geometry problems and real-life applications in fields like architecture and engineering.

Equations and Properties of Parallel and Perpendicular Lines

Slope from equation or points

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• Calculate slope using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ by plugging in the coordinates of two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line
• Identify the slope $m$ in the slope-intercept form of a line's equation [y = mx + b](https://www.fiveableKeyTerm:y_=_mx_+_b) where $b$ represents the y-intercept (point where the line crosses the y-axis)
• Determine the slope $m$ in the point-slope form of a line's equation $y - y_1 = m(x - x_1)$ using the coordinates of a single point $(x_1, y_1)$ on the line
• Find the slope of a line in general form [Ax + By + C](https://www.fiveableKeyTerm:ax_+_by_+_c) = 0 by rearranging the equation to solve for $m = -\frac{A}{B}$

Slopes of parallel vs perpendicular lines

• Recognize that parallel lines have identical slopes meaning if the slope of line 1 equals the slope of line 2 ($m_1 = m_2$) then the lines are parallel and will never intersect
• Identify perpendicular lines as having slopes that are negative reciprocals of each other satisfying the condition $m_1 \cdot m_2 = -1$ so if line 1 has a slope of $\frac{2}{3}$ then line 2 will have a slope of $-\frac{3}{2}$ for the lines to be perpendicular
• Understand that perpendicular lines intersect at a 90° angle while parallel lines maintain a constant distance between them without ever intersecting

Equations of parallel and perpendicular lines

• Write the equation of a line parallel to a given line passing through a specific point $(x_1, y_1)$ using the slope $m$ of the given line in slope-intercept form $y = m(x - x_1) + y_1$ or point-slope form $y - y_1 = m(x - x_1)$
• Determine the equation of a line perpendicular to a given line passing through a point $(x_1, y_1)$ by using the negative reciprocal of the given line's slope $-\frac{1}{m}$ in slope-intercept form $y = -\frac{1}{m}(x - x_1) + y_1$ or point-slope form $y - y_1 = -\frac{1}{m}(x - x_1)$
• Apply the properties of parallel and perpendicular lines to find missing information such as the slope or equation of a line given partial information about the lines and their relationships

Applications in coordinate geometry

• Compare the slopes of lines to determine if they are parallel ($m_1 = m_2$) or perpendicular ($m_1 \cdot m_2 = -1$) and use this information to solve geometric problems
• Calculate the distance between a point and a line by constructing a perpendicular line from the point to the given line and using the distance formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ between the point and the intersection of the perpendicular and given lines
• Solve problems involving triangles (right triangles, special triangles), quadrilaterals (parallelograms, rectangles, squares), and other polygons on the coordinate plane using the properties of parallel and perpendicular lines
• Find the point of intersection between two lines by setting their equations equal to each other and solving the resulting system of equations for the $x$ and $y$ coordinates of the point
• Utilize the concepts of parallel and perpendicular lines in real-world applications such as designing buildings (architecture), constructing roads (engineering), and plotting navigational routes (navigation) to solve practical problems