Question
perpendicular lines
Answers
GPT-4o mini
Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. In a Cartesian coordinate system, two lines can be represented by their equations, usually in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
For two lines to be perpendicular:
- The slopes of the lines must be negative reciprocals of each other. This means that if one line has a slope \(m_1\), and the other line has a slope \(m_2\), then the relationship can be expressed as:
\[ m_1 \times m_2 = -1 \]
- For example, if one line has a slope of 2 (i.e., \(m_1 = 2\)), the slope of the line perpendicular to it would be \(-\frac{1}{2}\) (i.e., \(m_2 = -\frac{1}{2}\)).
### Examples:
1. **Horizontal and Vertical Lines**: A horizontal line (slope = 0) is always perpendicular to a vertical line (undefined slope).
2. **Slope Example**:
- Line 1: \(y = 2x + 3\) (slope = 2)
- Line 2: \(y = -\frac{1}{2}x + 1\) (slope = \(-\frac{1}{2}\))
- These two lines are perpendicular.
### Graphical Representation:
When plotted on a graph, you can clearly see that perpendicular lines form a square corner where they intersect.
If you have specific questions about perpendicular lines or need examples or problems related to them, feel free to ask!
For two lines to be perpendicular:
- The slopes of the lines must be negative reciprocals of each other. This means that if one line has a slope \(m_1\), and the other line has a slope \(m_2\), then the relationship can be expressed as:
\[ m_1 \times m_2 = -1 \]
- For example, if one line has a slope of 2 (i.e., \(m_1 = 2\)), the slope of the line perpendicular to it would be \(-\frac{1}{2}\) (i.e., \(m_2 = -\frac{1}{2}\)).
### Examples:
1. **Horizontal and Vertical Lines**: A horizontal line (slope = 0) is always perpendicular to a vertical line (undefined slope).
2. **Slope Example**:
- Line 1: \(y = 2x + 3\) (slope = 2)
- Line 2: \(y = -\frac{1}{2}x + 1\) (slope = \(-\frac{1}{2}\))
- These two lines are perpendicular.
### Graphical Representation:
When plotted on a graph, you can clearly see that perpendicular lines form a square corner where they intersect.
If you have specific questions about perpendicular lines or need examples or problems related to them, feel free to ask!