No, when two lines have the same slope, they are not perpendicular; they are parallel.
Understanding the relationship between the slopes of lines is fundamental in geometry and algebra. The slope of a line defines its steepness and direction. Different slope relationships dictate how lines interact with each other on a plane.
Parallel Lines: Same Slope, Never Intersect
Lines that have the same slope are known as parallel lines. This means they run in the same direction and maintain a constant distance from each other, never intersecting, no matter how far they are extended.
- Key Characteristic: Equal slopes.
- Intersection: Never intersect.
- Example:
- Line 1:
y = 3x + 5(Slope = 3) - Line 2:
y = 3x - 2(Slope = 3)
Since both lines have a slope of3, they are parallel.
- Line 1:
Perpendicular Lines: Opposite Reciprocal Slopes
In contrast to parallel lines, perpendicular lines intersect at a perfect 90-degree angle. The relationship between their slopes is distinct and very specific. As per mathematical definition, the slopes of perpendicular lines are opposite reciprocals.
- Key Characteristic: Slopes are opposite reciprocals. If one line has a slope
m, a line perpendicular to it will have a slope of-1/m. - Intersection: Intersect at a right (90°) angle.
- Example:
- Line A:
y = 2x + 1(Slope = 2) - Line B:
y = -1/2x + 4(Slope = -1/2)
Here, the slope of Line B (-1/2) is the opposite reciprocal of the slope of Line A (2). This confirms they are perpendicular.
- Line A:
- Special Case: Horizontal and Vertical Lines:
- A horizontal line has a slope of
0. - A vertical line has an undefined slope.
- Horizontal and vertical lines are perpendicular to each other, as they also intersect at a 90-degree angle.
- A horizontal line has a slope of
Slope Relationships at a Glance
The table below summarizes the key differences between parallel and perpendicular lines based on their slopes:
| Relationship | Slope Condition | Intersection Type | Visual Representation |
|---|---|---|---|
| Parallel | Slopes are equal (m1 = m2) |
No Intersection | Two lines running alongside each other |
| Perpendicular | Slopes are opposite reciprocals (m1 * m2 = -1 or m2 = -1/m1) |
Right Angle (90°) | Two lines forming a perfect 'T' or '+' |
Understanding these fundamental slope relationships is crucial for various applications, from graphing equations to solving geometric problems. For more details on slopes, you can explore resources like Wikipedia's article on Slope.
