Parallel and Perpendicular Lines

TL;DR

Parallel lines have the exact same slope and never intersect, while perpendicular lines have slopes that are negative reciprocals of each other and intersect at a 90-degree angle. Understanding these slope relationships lets you write equations for new lines that are parallel or perpendicular to a given line. You can easily check if lines are parallel or perpendicular by comparing their slopes.

1. The Mental Model

Imagine two train tracks running side-by-side forever; that's parallel. Now imagine a perfect "plus sign" or a crossroad where roads meet at a perfect right angle; that's perpendicular. It all comes down to their slopes.

2. The Core Material

When we talk about lines on a coordinate plane, their slope tells us two key things: how steep they are and in what direction they're going. The beauty of parallel and perpendicular lines is that their slopes have very specific, predictable relationships.

Parallel Lines: Same Slope

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Two lines are parallel if they have the exact same slope and are different lines. Think of two roads running perfectly straight and never meeting. Since their slopes are identical, they'll always maintain the same distance from each other.

If line 1 has a slope of m1 and line 2 has a slope of m2:
For parallel lines: m1 = m2

Perpendicular Lines: Negative Reciprocal Slopes

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Two lines are perpendicular if they intersect at a perfect 90-degree angle. Their slopes have a special relationship: they are negative reciprocals of each other.

To find the negative reciprocal of a slope m:
1. Flip the fraction: If m is a/b, its reciprocal is b/a. If m is an integer like 3, think of it as 3/1, so its reciprocal is 1/3.
2. Change the sign: If the original slope was positive, the perpendicular slope is negative. If the original slope was negative, the perpendicular slope is positive.

So, if line 1 has a slope of m1 and line 2 has a slope of m2:
For perpendicular lines: m1 = -1/m2 (or m1 * m2 = -1)

Here's how to think about finding the slope of a perpendicular line:

graph TD
    A["Start with Original Slope (m)"] --> B{Is m defined?};
    B -- Yes --> C{Is m = 0?};
    C -- Yes --> D["Perpendicular Slope is undefined (Vertical Line)"];
    C -- No --> E{Is m undefined?};
    E -- Yes --> F["Perpendicular Slope is 0 (Horizontal Line)"];
    E -- No --> G["Flip the fraction (reciprocal)"];
    G --> H["Change the sign (negative)"];
    H --> I["Perpendicular Slope (-1/m)"];
    D --> End;
    F --> End;
    I --> End;
    End["Done"];

Vertical and Horizontal Lines

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• Horizontal lines have a slope of 0 (e.g., y = 3).
• Vertical lines have an undefined slope (e.g., x = -2).

A horizontal line is always perpendicular to a vertical line. This fits the negative reciprocal rule if you think of 0 as 0/1, so its negative reciprocal would be -1/0, which is undefined.

3. Worked Example

Let's say you're given a line with the equation y = (2/3)x - 5. You need to find the equation of:
1. A line parallel to it that passes through the point (6, 1).
2. A line perpendicular to it that also passes through the point (6, 1).

Step 1: Identify the slope of the given line.
The given equation y = (2/3)x - 5 is in slope-intercept form (y = mx + b).
So, the slope m is 2/3.

Part 1: Find the parallel line.
A parallel line will have the same slope: m_parallel = 2/3.
Now use the point-slope form y - y1 = m(x - x1) with the point (6, 1):
y - 1 = (2/3)(x - 6)
y - 1 = (2/3)x - (2/3)*6
y - 1 = (2/3)x - 4
y = (2/3)x - 4 + 1
y = (2/3)x - 3

Part 2: Find the perpendicular line.
The slope of the perpendicular line will be the negative reciprocal of 2/3.
Reciprocal of 2/3 is 3/2.
Negative reciprocal is -3/2. So, m_perpendicular = -3/2.
Now use the point-slope form y - y1 = m(x - x1) with the point (6, 1):
y - 1 = (-3/2)(x - 6)
y - 1 = (-3/2)x - (-3/2)*6
y - 1 = (-3/2)x + 9
y = (-3/2)x + 9 + 1
y = (-3/2)x + 10

So, the parallel line is y = (2/3)x - 3 and the perpendicular line is y = (-3/2)x + 10.

4. Key Takeaways

• Parallel lines have identical slopes. If m1 is the slope of line 1 and m2 is the slope of line 2, then m1 = m2.
• Perpendicular lines have slopes that are negative reciprocals. This means m1 * m2 = -1 (unless one slope is 0 and the other undefined).
• To find the negative reciprocal of a slope, flip the fraction and change its sign.
• Horizontal lines (slope 0) are perpendicular to vertical lines (undefined slope).
• Always ensure your line equations are in slope-intercept form (y = mx + b) or you can easily convert them to find the slope m.

Common Mistakes to Avoid:
- Forgetting the negative sign when finding a perpendicular slope; just flipping the fraction isn't enough.
- Not flipping the fraction for perpendicular lines; simply changing the sign is incorrect.
- Assuming two lines are parallel just because their y-intercepts are different; their slopes must be the same.
- Thinking a line is perpendicular if slopes are just reciprocals, ignoring the negative sign requirement.
- Not handling vertical/horizontal lines correctly; a slope of 0 means the perpendicular slope is undefined, not 0.

5. Now Try It

You're given a line with the equation 3x + 4y = 12.
1. Find the equation of a line parallel to it that passes through the point (-2, 5).
2. Find the equation of a line perpendicular to it that also passes through the point (-2, 5).

For both, express your final answer in slope-intercept form (y = mx + b).

Success looks like: You correctly convert the given equation to find its slope, then apply the correct slope relationships for parallel and perpendicular lines, and finally use the point-slope form to get the final equations passing through the specified point.