๐ Understanding Perpendicular Lines
In geometry, two lines are perpendicular if they intersect at a right angle (90 degrees). Understanding this concept is fundamental to solving problems involving perpendicular line equations, especially when using the point-slope form. This form helps us define a line using a single point and the slope of the line.
๐ Historical Context
The study of perpendicularity dates back to ancient Greece, with mathematicians like Euclid laying the foundations of geometry. The concept of slope, however, developed later with the advent of coordinate geometry by Renรฉ Descartes in the 17th century. The combination of these ideas gives us the tools to easily define and manipulate lines using equations.
๐ Key Principles: Point-Slope Form and Slopes of Perpendicular Lines
The point-slope form of a linear equation is given by:
$y - y_1 = m(x - x_1)$
where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. The crucial aspect when dealing with perpendicular lines is understanding the relationship between their slopes.
- ๐ Negative Reciprocal: If a line has a slope of $m$, a line perpendicular to it will have a slope of $-\frac{1}{m}$. This means you flip the fraction and change the sign.
- ๐ Point on the Line: You need a point $(x_1, y_1)$ that the perpendicular line passes through. This point, along with the new slope, goes directly into the point-slope form.
- โ๏ธ Equation Substitution: Substitute the new slope and the given point into the point-slope formula $y - y_1 = m(x - x_1)$.
โ๏ธ Practical Examples
Example 1:
Find the equation of a line perpendicular to $y = 2x + 3$ and passing through the point $(1, 5)$.
- Identify the Slope: The slope of the given line is $m = 2$.
- Find the Perpendicular Slope: The slope of the perpendicular line is $-\frac{1}{2}$.
- Apply Point-Slope Form: Using the point $(1, 5)$ and the slope $-\frac{1}{2}$, the equation is:
$y - 5 = -\frac{1}{2}(x - 1)$
Example 2:
Find the equation of a line perpendicular to $y = -\frac{3}{4}x - 1$ and passing through the point $(-2, 3)$.
- Identify the Slope: The slope of the given line is $m = -\frac{3}{4}$.
- Find the Perpendicular Slope: The slope of the perpendicular line is $\frac{4}{3}$.
- Apply Point-Slope Form: Using the point $(-2, 3)$ and the slope $\frac{4}{3}$, the equation is:
$y - 3 = \frac{4}{3}(x + 2)$
๐ก Tips for Success
- โ๏ธ Double-Check: Always double-check that you've correctly calculated the negative reciprocal of the original slope.
- ๐งญ Visualization: Try to visualize the lines on a graph to ensure they appear perpendicular.
- ๐ข Simplification: While the point-slope form is perfectly valid, you might be asked to convert it to slope-intercept form ($y = mx + b$) or standard form ($Ax + By = C$). Practice these conversions.
๐ Conclusion
Understanding perpendicular lines and using the point-slope form is a key skill in algebra and geometry. By mastering the concept of negative reciprocals and practicing applying the point-slope formula, you can confidently solve these types of problems.