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Parallel and Perpendicular Lines: Finding Equations
In geometry, understanding the relationships between lines is fundamental. Two common relationships are parallelism and perpendicularity. Parallel lines never intersect, while perpendicular lines intersect at a right angle. This lesson explores how to find the equations of lines that are parallel or perpendicular to a given line and pass through a specific point.
Parallel Lines
Parallel lines have the same slope. This means that if you have the equation of one line, you can easily find the equation of a parallel line by keeping the slope the same and adjusting the y-intercept.
Example 1:
Find the equation of the line that is parallel to the line y = 2x + 3 and passes through the point (1, 5).
**Solution:**
- The given line has a slope of 2 (the coefficient of x).
- Since we want a parallel line, the new line will also have a slope of 2.
- We can use the point-slope form of a linear equation to find the equation of the new line:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point.
Substituting the values, we get:
y - 5 = 2(x - 1)
Simplifying the equation, we get:
y = 2x + 3
Therefore, the equation of the line parallel to y = 2x + 3 and passing through (1, 5) is y = 2x + 3.
Perpendicular Lines
Perpendicular lines have slopes that are negative reciprocals of each other. This means that if the slope of one line is m, the slope of a perpendicular line is -1/m.
Example 2:
Find the equation of the line that is perpendicular to the line y = -3x + 1 and passes through the point (2, 4).
**Solution:**
- The given line has a slope of -3.
- The slope of a perpendicular line is the negative reciprocal of -3, which is 1/3.
- Using the point-slope form, we get:
y - 4 = (1/3)(x - 2)
Simplifying, we get:
y = (1/3)x + 10/3
Therefore, the equation of the line perpendicular to y = -3x + 1 and passing through (2, 4) is y = (1/3)x + 10/3.
Key Takeaways
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals of each other.
- The point-slope form of a linear equation (y - y1 = m(x - x1)) is helpful for finding the equation of a line when you know the slope and a point on the line.
By understanding these concepts, you can effectively find the equations of lines that are parallel or perpendicular to a given line, which is a valuable skill in various mathematical applications.