**Title: Parallel and Perpendicular Lines – Definitions, Examples & Importance**
Understanding the concepts of parallel and perpendicular lines is fundamental in geometry. These concepts play a crucial role in designing buildings, roadways, art, and even scientific research.
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## What are Parallel and Perpendicular Lines?
**Parallel lines** are lines in a plane that never meet. They are always the same distance apart and have the same slope.
**Perpendicular lines** are lines that intersect at a right angle (90 degrees). Their slopes are negative reciprocals of each other.
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##
Parallel Lines
Parallel lines never intersect and remain equidistant, having the same slope. They are denoted using the symbol \( \parallel \).
### Equation Form:
Two lines \( L_1: y = m_1x + c_1 \) and \( L_2: y = m_2x + c_2 \) are parallel if
\[
m_1 = m_2
\]
—
### Easy Questions:
**Q1:** Are the lines \( y = 2x + 3 \) and \( y = 2x – 5 \) parallel?
**Solution:** Both lines have the same slope \( m = 2 \), so they are parallel.
**Q2:** Find the slope of the line parallel to \( y = 5x + 4 \).
**Solution:** The slope is \( m = 5 \), so any line with slope 5 is parallel.
—
### Medium Questions:
**Q1:** Find if the lines \( 3x – y = 7 \) and \( 6x – 2y = 8 \) are parallel.
**Solution:**
Convert both to slope-intercept form:
First, \( y = 3x – 7 \), slope is 3.
Second: \( 2y = 6x – 8 \Rightarrow y = 3x – 4 \).
Slopes are same ⇒ Parallel.
**Q2:** Find a line parallel to \( y = -\frac{1}{2}x + 1 \) passing through (0, -2).
**Solution:**
Use point-slope form:
\[
y + 2 = -\frac{1}{2}x \Rightarrow y = -\frac{1}{2}x – 2
\]
—
### Hard Questions:
**Q1:** Prove that the lines passing through points (1, 2), (3, 6) and (-2, -1), (0, 3) are parallel.
**Solution:**
Slope of first: \( m = \frac{6-2}{3-1} = 2 \)
Slope of second: \( m = \frac{3-(-1)}{0 – (-2)} = \frac{4}{2} = 2 \)
Since both slopes are equal ⇒ Parallel
**Q2:** Find the equation of a line parallel to \( 4x – 5y = 2 \) and passing through point (1, -1).
**Solution:**
Convert to slope-intercept: \( y = \frac{4}{5}x – \frac{2}{5} \)
Use point-slope form:
\[
y + 1 = \frac{4}{5}(x – 1) \Rightarrow y = \frac{4}{5}x – \frac{9}{5}
\]
—
##
Perpendicular Lines
Perpendicular lines intersect at a right angle, and their slopes are negative reciprocals. If \( m_1 \cdot m_2 = -1 \), the lines are perpendicular.
—
### Easy Questions:
**Q1:** Is the line \( y = 3x + 4 \) perpendicular to \( y = -\frac{1}{3}x – 5 \)?
**Solution:**
Multiplying slopes: \( 3 \cdot (-\frac{1}{3}) = -1 \), so Yes.
**Q2:** What is the slope of a line perpendicular to slope 4?
**Solution:**
Slope = \( -\frac{1}{4} \)
—
### Medium Questions:
**Q1:** Check if lines \( y = 2x + 1 \) and \( y = -\frac{1}{2}x + 4 \) are perpendicular.
**Solution:**
Product of slopes: \( 2 \cdot (-\frac{1}{2}) = -1 \) ⇒ Perpendicular
**Q2:** Find equation of a line perpendicular to \( y = \frac{2}{3}x – 1 \) through point (0, 4).
**Solution:**
Slope = \( -\frac{3}{2} \)
\[
y – 4 = -\frac{3}{2}(x – 0) \Rightarrow y = -\frac{3}{2}x + 4
\]
—
### Hard Questions:
**Q1:** Show that lines passing through (2, 1), (4, 5) and (3, 7), (5, 6) are perpendicular.
**Solution:**
Slope 1: \( \frac{5 – 1}{4 – 2} = 2 \)
Slope 2: \( \frac{6 – 7}{5 – 3} = \frac{-1}{2} \)
Product = \( 2 \cdot -\frac{1}{2} = -1 \) ⇒ Perpendicular
**Q2:** Find equation of line perpendicular to \( 5x + y = 10 \), and goes through (2,3).
**Solution:**
Slope of given line: \( y = -5x + 10 \), slope = -5
Perpendicular slope = \( \frac{1}{5} \)
\[
y – 3 = \frac{1}{5}(x – 2) \Rightarrow y = \frac{1}{5}x + \frac{13}{5}
\]
—
##
Difference Between Parallel and Perpendicular Lines
Parallel lines never meet; perpendicular lines intersect at 90°. Slopes are equal in parallel and negative reciprocals in perpendicular lines.
### Summary Table:
| Type | Condition for Slopes | Intersection |
|————–|——————————-|————–|
| Parallel | \( m_1 = m_2 \) | Never |
| Perpendicular| \( m_1 \cdot m_2 = -1 \) | Right angle |
—
### Easy Questions:
**Q1:** What is the difference between parallel and perpendicular slopes?
**Solution:**
Parallel: same slope. Perpendicular: negative reciprocals.
**Q2:** Can two perpendicular lines be parallel?
**Solution:**
No, perpendicular lines must intersect at 90°.
—
### Medium Questions:
**Q1:** Determine whether lines \( y = -2x + 3 \) and \( y = 2x + 1 \) are parallel or perpendicular.
**Solution:**
Product \( -2 \cdot 2 = -4 \). Neither parallel nor perpendicular.
**Q2:** Are lines with slopes \( \frac{1}{3} \) and \( -\frac{3}{1} \) perpendicular?
**Solution:**
\( \frac{1}{3} \cdot -3 = -1 \) ⇒ Yes, perpendicular
—
### Hard Questions:
**Q1:** Find the type of relation between lines \( 7x – 2y = 10 \) and \( 14x – 4y = -3 \).
**Solution:**
Convert both to slope-intercept form and compare:
Both have same slope ⇒ Parallel
**Q2:** Line A has slope \( 4 \). Line B is perpendicular. What is the slope of a line C parallel to line B?
**Solution:**
Line B slope = \( -\frac{1}{4} \) ⇒ Line C slope = \( -\frac{1}{4} \)
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## Why Is It Important to Study Parallel and Perpendicular Lines?
Learning about parallel and perpendicular lines enhances your understanding of geometry, engineering, and architecture. These fundamental concepts are foundational for trigonometry and coordinate geometry.
They also form the backbone of real-world structures, navigation systems, machine learning visualizations, and design processes.
### Recent Contributions by Mathematicians
Mathematicians like Terence Tao and Maryam Mirzakhani (posthumously) have contributed significantly in geometry and spatial reasoning, showing the importance of understanding plane relationships like parallelism and perpendicularity, especially in non-Euclidean spaces and topology.
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