3.4 Equations of parallel and perpendicular lines

Equations and Properties of Parallel and Perpendicular Lines

Understanding how parallel and perpendicular lines behave on the coordinate plane lets you move between geometric relationships and algebra. If you can find a slope, you can determine whether two lines are parallel, perpendicular, or neither, and then write equations for new lines that satisfy those relationships.

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Slope from an Equation or Two Points

From two points: Use the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Plug in the coordinates of any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. For example, given $(1, 3)$ and $(4, 9)$, the slope is $m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$.

From an equation: The method depends on the form you're given.

• Slope-intercept form $y = mx + b$: The slope is $m$ and the y-intercept is $b$. You can read the slope directly.
• Point-slope form $y - y_1 = m(x - x_1)$: The slope is the coefficient $m$ out front. The point $(x_1, y_1)$ is built into the equation.
• Standard/general form $Ax + By + C = 0$: Rearrange to slope-intercept form by solving for $y$. The slope works out to $m = -\frac{A}{B}$. For instance, $3x + 2y - 6 = 0$ gives $y = -\frac{3}{2}x + 3$, so $m = -\frac{3}{2}$.

Slopes of Parallel vs. Perpendicular Lines

Parallel lines have the same slope:

$m_1 = m_2$

Because they rise and run at the same rate, they never intersect and stay a constant distance apart. For example, $y = 4x + 1$ and $y = 4x - 7$ are parallel since both have slope $4$.

Perpendicular lines have slopes that are negative reciprocals of each other:

$m_1 \cdot m_2 = -1$

This means you flip the fraction and change the sign. If one line has slope $\frac{2}{3}$, a line perpendicular to it has slope $-\frac{3}{2}$. Perpendicular lines intersect at exactly 90°.

Writing Equations of Parallel and Perpendicular Lines

The process is the same for both cases. The only difference is which slope you use.

To write a line parallel to a given line through a point $(x_1, y_1)$:

1. Find the slope $m$ of the given line.

2. Use that same slope $m$ (parallel lines share slopes).

3. Plug into point-slope form: $y - y_1 = m(x - x_1)$.

4. Simplify to slope-intercept form if the problem asks for it.

Example: Write the equation of a line parallel to $y = 3x - 5$ through $(2, 1)$.

The given slope is $3$. Using point-slope form: $y - 1 = 3(x - 2)$, which simplifies to $y = 3x - 5$... wait, that's the same line. Actually: $y - 1 = 3(x - 2) \rightarrow y = 3x - 6 + 1 \rightarrow y = 3x - 5$. In this case the point $(2,1)$ is on the original line. If the point were $(2, 4)$ instead: $y - 4 = 3(x - 2) \rightarrow y = 3x - 2$.

To write a line perpendicular to a given line through a point $(x_1, y_1)$:

1. Find the slope $m$ of the given line.

2. Take the negative reciprocal: the new slope is $-\frac{1}{m}$.

3. Plug into point-slope form: $y - y_1 = -\frac{1}{m}(x - x_1)$.

4. Simplify as needed.

Example: Write the equation of a line perpendicular to $y = \frac{2}{3}x + 4$ through $(6, -1)$. The given slope is $\frac{2}{3}$, so the perpendicular slope is $-\frac{3}{2}$. Using point-slope form: $y - (-1) = -\frac{3}{2}(x - 6) \rightarrow y + 1 = -\frac{3}{2}x + 9 \rightarrow y = -\frac{3}{2}x + 8$.

Applications in Coordinate Geometry

These slope relationships show up constantly when you work with shapes on the coordinate plane.

• Classifying lines: Compare slopes of two lines to determine if they're parallel ($m_1 = m_2$), perpendicular ($m_1 \cdot m_2 = -1$), or neither.
• Verifying shapes: To prove a quadrilateral is a rectangle, show that opposite sides are parallel (equal slopes) and consecutive sides are perpendicular (negative reciprocal slopes). Similar reasoning applies to parallelograms, squares, and right triangles.
• Finding intersections: Set two line equations equal to each other and solve the system for $x$ and $y$. That gives you the point where the lines cross.
• Distance from a point to a line: Construct a perpendicular from the point to the given line, find where that perpendicular intersects the line, then use the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ between the original point and the intersection point.

These tools come up in applied contexts too: verifying that walls meet at right angles in architecture, confirming road segments are parallel in engineering, or checking alignment in design work. But for this course, the main payoff is being able to move fluently between slopes, equations, and geometric relationships on the coordinate plane.