Consider The Line 5x 8y= 3. What Is The Slope Of A Line Perpendicular To This Line? What Is The Slope Of
Understanding the slope of lines is fundamental in coordinate geometry, offering insights into the direction and angle of lines on the Cartesian plane. When analyzing the equation of a line such as 5x + 8y = 3, one of the key questions often posed is: What is the slope of a line perpendicular to this line? This article provides a comprehensive exploration of this question, including how to determine the slope of the given line, the concept of perpendicular slopes, and practical applications. Whether you're a student preparing for exams or someone interested in mathematical concepts, this guide offers valuable insights into slope calculations and their significance in geometry.
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Understanding the Equation of the Line 5x + 8y = 3
Converting to Slope-Intercept Form
To analyze the slope of a line, it's most straightforward to convert the given equation into slope-intercept form, which is:\[ y = mx + b \]
where:
- \( m \) is the slope
- \( b \) is the y-intercept
Let's convert the equation \( 5x + 8y = 3 \):
- Subtract \( 5x \) from both sides:
- Divide both sides by 8:
So, the slope \( m \) of the given line is:
\[ m = -\frac{5}{8} \]
Key Point: The slope of the line \( 5x + 8y = 3 \) is \( -\frac{5}{8} \).
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What Is The Slope Of A Line Perpendicular To This Line?
Understanding Perpendicular Lines
Perpendicular lines are lines that intersect at a right angle (90 degrees). A critical property of perpendicular lines in the Cartesian plane is the relationship between their slopes:- If a line has a slope \( m \), then the slope of any line perpendicular to it is the negative reciprocal of \( m \).
\[ m_1 = m \]
then the slope of the perpendicular line \( m_2 \) is:
\[ m_2 = -\frac{1}{m} \]
Note: This only applies when \( m \neq 0 \). For vertical lines (which have undefined slopes), the perpendicular line is horizontal, and vice versa.
Calculating the Perpendicular Slope
Given the slope of the original line:\[ m = -\frac{5}{8} \]
The slope of the line perpendicular to it is:
\[ m_{\perp} = -\frac{1}{-\frac{5}{8}} = \frac{8}{5} \]
Conclusion: The slope of a line perpendicular to \( 5x + 8y = 3 \) is \( \frac{8}{5} \).
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Why Is The Perpendicular Slope Important?
Understanding the slope of perpendicular lines has practical applications across various fields:
- Geometry & Trigonometry: Helps in calculating angles between lines.
- Engineering & Architecture: Critical for designing structures with right angles.
- Computer Graphics: Used in algorithms for rendering shapes and patterns.
- Physics: Important in analyzing vectors and forces at right angles.
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How To Find The Equation of a Perpendicular Line
Suppose you want to find the equation of a line perpendicular to \( 5x + 8y=3 \) passing through a specific point, say \( (x0, y0) \). Here’s a step-by-step approach:
Step 1: Identify the Slope
- From earlier, the slope of the original line is \( -\frac{5}{8} \).
- The perpendicular slope is \( \frac{8}{5} \).
Step 2: Use Point-Slope Form
The point-slope form of a line is:\[ y - y0 = m (x - x0) \]
- Plug in the point \( (x0, y0) \) and the perpendicular slope \( m = \frac{8}{5} \).
Step 3: Write the Equation
For example, if the point is \( (2, 3) \):
\[ y - 3 = \frac{8}{5}(x - 2) \]
This can be simplified to slope-intercept form or standard form as needed.
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Additional Concepts Related to Slopes and Perpendicular Lines
1. Parallel Lines
- Lines that never intersect and have the same slope.
- For the given line \( y = -\frac{5}{8}x + \frac{3}{8} \), any line parallel to it will also have a slope of \( -\frac{5}{8} \).
2. Vertical and Horizontal Lines
- Vertical lines have undefined slopes and are expressed as \( x = c \).
- Horizontal lines have a slope of zero, expressed as \( y = c \).
- The perpendicular to a vertical line is horizontal, and vice versa.
3. Slope and Angle Relationship
- The angle \( \theta \) that a line makes with the positive x-axis is related to its slope \( m \) by:
- For the original line:
- For the perpendicular line:
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Practical Applications and Example Problems
Example 1: Find the Equation of a Perpendicular Line
Problem: Find the equation of the line perpendicular to \( 5x + 8y = 3 \) passing through \( (4, 2) \).Solution:
- The perpendicular slope:
\[ m_{\perp} = \frac{8}{5} \]
- Use point-slope form:
\[ y - 2 = \frac{8}{5}(x - 4) \]
- Simplify:
\[ y - 2 = \frac{8}{5}x - \frac{32}{5} \]
\[ y = \frac{8}{5}x - \frac{32}{5} + 2 \]
\[ y = \frac{8}{5}x - \frac{32}{5} + \frac{10}{5} \]
\[ y = \frac{8}{5}x - \frac{22}{5} \]
Answer: The equation of the perpendicular line is:
\[ y = \frac{8}{5}x - \frac{22}{5} \]
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Summary and Key Takeaways
- The original line \( 5x + 8y = 3 \) has a slope of \( -\frac{5}{8} \).
- The slope of a line perpendicular to this line is \( \frac{8}{5} \).
- Perpendicular lines have slopes that are negative reciprocals.
- To find the equation of a perpendicular line passing through a point, use the point-slope form with the perpendicular slope.
- Understanding slopes and perpendicularity is essential in geometry, engineering, physics, and computer science.
Final Thoughts
Mastering the concept of slopes, especially in the context of perpendicular lines, enhances your ability to analyze geometric relationships and solve complex problems. Remember, the key to understanding slopes lies in converting equations into slope-intercept form, recognizing reciprocal relationships, and applying the principles to real-world scenarios. Keep practicing with different equations and points to strengthen your grasp of these fundamental concepts in coordinate geometry.
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