Unit 3 Parallel And Perpendicular Lines Homework 3 Answer Key

Navigating the world of geometry can often feel like traversing a complex maze, particularly when dealing with concepts like parallel and perpendicular lines. Understanding these fundamental relationships is crucial, not just for academic success but also for real-world applications in fields like architecture, engineering, and even everyday problem-solving. This comprehensive guide delves into the intricacies of Unit 3 Homework 3, providing a clear and concise answer key alongside explanations that illuminate the underlying principles.

Unveiling the Fundamentals: Parallel and Perpendicular Lines

Before diving into the specific problems, it's essential to establish a solid understanding of what parallel and perpendicular lines are. Parallel lines are lines in a plane that never intersect, maintaining a constant distance from each other. A key characteristic of parallel lines is that they have the same slope. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees). The slopes of perpendicular lines are negative reciprocals of each other. Understanding these definitions is the bedrock for tackling any geometry problem involving these lines.

Unit 3 Homework 3: A Problem-Solving Journey

Let's embark on a detailed exploration of the problems within Unit 3 Homework 3, complete with explanations and answer keys to guide you through each step.

Problem 1: Identifying Parallel and Perpendicular Lines from Equations

The Problem: Determine whether the given pairs of lines are parallel, perpendicular, or neither.

• Line 1: y = 2x + 3
• Line 2: y = 2x - 1

Solution:

• Step 1: Identify the slopes. The slope of a line in the form y = mx + b is 'm'.

  • Line 1 has a slope of 2.
  • Line 2 has a slope of 2.

• Step 2: Compare the slopes.

  • Since the slopes are equal (2 = 2), the lines are parallel.

Answer: Parallel

Problem 2: Identifying Parallel and Perpendicular Lines from Equations

The Problem: Determine whether the given pairs of lines are parallel, perpendicular, or neither.

• Line 1: y = -1/3x + 5
• Line 2: y = 3x - 2

Solution:

• Step 1: Identify the slopes.

  • Line 1 has a slope of -1/3.
  • Line 2 has a slope of 3.

• Step 2: Determine if the slopes are negative reciprocals. The negative reciprocal of -1/3 is 3 (because -1/3 * 3 = -1).

  • Since the slopes are negative reciprocals of each other, the lines are perpendicular.

Answer: Perpendicular

Problem 3: Identifying Parallel and Perpendicular Lines from Equations

The Problem: Determine whether the given pairs of lines are parallel, perpendicular, or neither.

• Line 1: y = 4x + 1
• Line 2: y = -4x + 6

Solution:

• Step 1: Identify the slopes.

  • Line 1 has a slope of 4.
  • Line 2 has a slope of -4.

• Step 2: Compare the slopes.

  • The slopes are not equal (4 ≠ -4), so the lines are not parallel.
  • The slopes are not negative reciprocals of each other (4 * -4 ≠ -1), so the lines are not perpendicular.

Answer: Neither

Problem 4: Writing the Equation of a Line Parallel to a Given Line

The Problem: Write the equation of a line that is parallel to y = 5x - 2 and passes through the point (1, 3).

Solution:

• Step 1: Identify the slope of the given line. The slope of y = 5x - 2 is 5.
• Step 2: Use the same slope for the parallel line. A parallel line will also have a slope of 5.
• Step 3: Use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.
  • y - 3 = 5(x - 1)
• Step 4: Simplify the equation to slope-intercept form (y = mx + b).
  • y - 3 = 5x - 5
  • y = 5x - 5 + 3
  • y = 5x - 2

Answer: y = 5x - 2

Problem 5: Writing the Equation of a Line Perpendicular to a Given Line

The Problem: Write the equation of a line that is perpendicular to y = -2x + 4 and passes through the point (2, -1).

Solution:

• Step 1: Identify the slope of the given line. The slope of y = -2x + 4 is -2.
• Step 2: Find the negative reciprocal of the slope. The negative reciprocal of -2 is 1/2.
• Step 3: Use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.
  • y - (-1) = 1/2(x - 2)
  • y + 1 = 1/2(x - 2)
• Step 4: Simplify the equation to slope-intercept form (y = mx + b).
  • y + 1 = 1/2x - 1
  • y = 1/2x - 1 - 1
  • y = 1/2x - 2

Answer: y = 1/2x - 2

Problem 6: Determining Parallelism and Perpendicularity from Two Points on Each Line

The Problem: Line 1 passes through points (1, 5) and (3, 9). Line 2 passes through points (-1, 6) and (1, 2). Are the lines parallel, perpendicular, or neither?

Solution:

• Step 1: Calculate the slope of Line 1. Use the formula m = (y2 - y1) / (x2 - x1).
  • m1 = (9 - 5) / (3 - 1) = 4 / 2 = 2
• Step 2: Calculate the slope of Line 2.
  • m2 = (2 - 6) / (1 - (-1)) = -4 / 2 = -2
• Step 3: Compare the slopes.
  • The slopes are not equal (2 ≠ -2), so the lines are not parallel.
  • The slopes are not negative reciprocals of each other (2 * -2 ≠ -1), so the lines are not perpendicular.

Answer: Neither

Problem 7: Determining Parallelism and Perpendicularity from Two Points on Each Line

The Problem: Line 1 passes through points (2, 4) and (4, 8). Line 2 passes through points (1, 3) and (5, 11). Are the lines parallel, perpendicular, or neither?

Solution:

• Step 1: Calculate the slope of Line 1.
  • m1 = (8 - 4) / (4 - 2) = 4 / 2 = 2
• Step 2: Calculate the slope of Line 2.
  • m2 = (11 - 3) / (5 - 1) = 8 / 4 = 2
• Step 3: Compare the slopes.
  • Since the slopes are equal (2 = 2), the lines are parallel.

Answer: Parallel

Problem 8: Determining Parallelism and Perpendicularity from Two Points on Each Line

The Problem: Line 1 passes through points (-2, 3) and (1, 5). Line 2 passes through points (0, 2) and (2, -1). Are the lines parallel, perpendicular, or neither?

Solution:

• Step 1: Calculate the slope of Line 1.
  • m1 = (5 - 3) / (1 - (-2)) = 2 / 3
• Step 2: Calculate the slope of Line 2.
  • m2 = (-1 - 2) / (2 - 0) = -3 / 2
• Step 3: Compare the slopes.
  • The slopes are not equal (2/3 ≠ -3/2), so the lines are not parallel.
  • The slopes are negative reciprocals of each other (2/3 * -3/2 = -1), so the lines are perpendicular.

Answer: Perpendicular

Problem 9: Application of Parallel and Perpendicular Lines in Geometry

The Problem: A triangle has vertices A(1, 2), B(4, 4), and C(2, 6). Determine if the triangle is a right triangle.

Solution:

• Step 1: Calculate the slopes of the sides of the triangle.
  • Slope of AB: mAB = (4 - 2) / (4 - 1) = 2 / 3
  • Slope of BC: mBC = (6 - 4) / (2 - 4) = 2 / -2 = -1
  • Slope of AC: mAC = (6 - 2) / (2 - 1) = 4 / 1 = 4
• Step 2: Check if any two sides are perpendicular.
  • Check if mAB * mBC = -1: (2/3) * (-1) = -2/3 ≠ -1
  • Check if mAB * mAC = -1: (2/3) * (4) = 8/3 ≠ -1
  • Check if mBC * mAC = -1: (-1) * (4) = -4 ≠ -1
• Step 3: Conclude based on the calculations.
  • Since no two sides are perpendicular, the triangle is not a right triangle.

Answer: Not a right triangle

Problem 10: Applying Parallel and Perpendicular Lines in Real-World Contexts

The Problem: A city planner is designing a new street grid. Street A is defined by the equation y = -3/4x + 2. Street B needs to be perpendicular to Street A and pass through the point (3, 1). Find the equation for Street B.

Solution:

• Step 1: Identify the slope of Street A. The slope of y = -3/4x + 2 is -3/4.
• Step 2: Find the negative reciprocal of the slope. The negative reciprocal of -3/4 is 4/3.
• Step 3: Use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.
  • y - 1 = 4/3(x - 3)
• Step 4: Simplify the equation to slope-intercept form (y = mx + b).
  • y - 1 = 4/3x - 4
  • y = 4/3x - 4 + 1
  • y = 4/3x - 3

Answer: y = 4/3x - 3

The Underlying Mathematics: A Deeper Dive

To truly master parallel and perpendicular lines, it's crucial to understand the underlying mathematical principles.

• Slope: The slope of a line, often denoted as 'm', represents the steepness and direction of the line. It's calculated as the change in y divided by the change in x (rise over run). The formula is m = (y2 - y1) / (x2 - x1).
• Slope-Intercept Form: The slope-intercept form of a linear equation is y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis).
• Point-Slope Form: The point-slope form of a linear equation is y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. This form is particularly useful when you know a point on the line and its slope.
• Parallel Lines: Parallel lines have the same slope. This means that their steepness and direction are identical, ensuring they never intersect.
• Perpendicular Lines: Perpendicular lines intersect at a right angle (90 degrees). The slopes of perpendicular lines are negative reciprocals of each other. If one line has a slope of 'm', the slope of a line perpendicular to it is '-1/m'. The product of their slopes is always -1.

Common Pitfalls and How to Avoid Them

Working with parallel and perpendicular lines can sometimes lead to common errors. Here are a few pitfalls to watch out for:

• Confusing Negative Reciprocals: A common mistake is to only negate the slope or only find the reciprocal, but not both. Remember to both negate and reciprocate to find the slope of a perpendicular line.
• Incorrectly Calculating Slope: When given two points, ensure you subtract the y-coordinates and x-coordinates in the correct order. Double-check your calculations to avoid errors.
• Forgetting to Simplify: After using the point-slope form, remember to simplify the equation into slope-intercept form (y = mx + b) or standard form (Ax + By = C), as required by the problem.
• Misinterpreting "Neither": If lines are neither parallel nor perpendicular, it simply means their slopes don't meet the criteria for either relationship. Don't assume there's always a relationship.

Real-World Applications: Seeing Geometry in Action

The concepts of parallel and perpendicular lines extend far beyond the classroom. Here are some real-world examples:

• Architecture: Architects use parallel and perpendicular lines extensively in building design. Walls are often parallel to each other, and perpendicular lines are crucial for ensuring corners are square and structures are stable.
• Engineering: Engineers rely on these principles for designing bridges, roads, and other infrastructure. Parallel lines ensure lanes of traffic remain separate, while perpendicular lines are essential for structural support.
• Navigation: Maps and navigation systems use coordinate systems based on parallel and perpendicular lines. Latitude and longitude lines, for example, form a grid that helps determine locations.
• Computer Graphics: In computer graphics, parallel and perpendicular lines are used to create 2D and 3D images. They are fundamental to defining shapes, perspectives, and spatial relationships.
• Everyday Life: From arranging furniture in a room to parking a car, understanding parallel and perpendicular lines helps us navigate and organize our physical environment.

FAQs: Addressing Common Questions

• Q: How can I quickly determine if two lines are parallel?

  • A: Check if their slopes are equal. If the slopes are the same, the lines are parallel.

• Q: What does it mean if the slopes of two lines multiply to -1?

  • A: It means the lines are perpendicular.

• Q: Can a vertical line and a horizontal line be perpendicular?

  • A: Yes, a vertical line has an undefined slope, and a horizontal line has a slope of 0. They intersect at a right angle and are therefore perpendicular.

• Q: Is it possible for a line to be both parallel and perpendicular to another line?

  • A: No, a line cannot be both parallel and perpendicular to another line simultaneously. These relationships are mutually exclusive.

• Q: Why are negative reciprocals important for perpendicular lines?

  • A: Negative reciprocals ensure that the lines intersect at a right angle. This relationship is mathematically proven and essential for geometric accuracy.

Conclusion: Mastering the Art of Lines

Parallel and perpendicular lines are fundamental concepts in geometry with far-reaching applications. By understanding the definitions, mastering the formulas, and practicing problem-solving, you can confidently navigate the challenges presented in Unit 3 Homework 3 and beyond. This comprehensive guide provides the necessary tools and knowledge to unlock the secrets of these essential geometric relationships. Embrace the challenge, explore the possibilities, and watch your understanding of geometry soar.