Unit 3 Parallel And Perpendicular Lines Homework 6 Answer Key

Unit 3: Parallel and Perpendicular Lines - Homework 6 Answer Key: A Comprehensive Guide

This comprehensive guide provides detailed solutions and explanations for Homework 6 in Unit 3, focusing on parallel and perpendicular lines. We'll delve into various concepts, including slopes, equations of lines, and geometric properties, ensuring a thorough understanding of the topic. This guide is designed to be helpful for students of all levels, from those seeking quick answers to those striving for deeper comprehension.

Understanding Parallel and Perpendicular Lines

Before diving into the solutions, let's solidify our understanding of the core concepts:

Parallel Lines:

Parallel lines are lines in a plane that never intersect. A key characteristic is that they have the same slope. If two lines have equations in the slope-intercept form (y = mx + b), where 'm' represents the slope, then they are parallel if their 'm' values are identical.

Example: The lines y = 2x + 3 and y = 2x - 5 are parallel because both have a slope of 2.

Perpendicular Lines:

Perpendicular lines intersect at a right angle (90 degrees). Their slopes have a specific relationship: they are negative reciprocals of each other. This means that if one line has a slope 'm', the slope of a line perpendicular to it is -1/m.

Example: The line y = 3x + 1 is perpendicular to the line y = (-1/3)x + 2. The slope of the first line is 3, and the slope of the second line is -1/3 (the negative reciprocal of 3).

Exception: Vertical and horizontal lines are perpendicular. A vertical line has an undefined slope, and a horizontal line has a slope of 0.

Homework 6 Problems and Solutions

We'll now tackle potential problems from Homework 6, providing detailed solutions and explanations. Remember, the specific problems in your homework might vary; however, the principles and methods illustrated here are universally applicable.

Problem 1: Finding the Slope of Parallel and Perpendicular Lines

Question: Given the line y = 4x - 7, find: a) The slope of a line parallel to the given line. b) The slope of a line perpendicular to the given line.

Solution:

a) Parallel Line Slope: Parallel lines have the same slope. The given line has a slope of 4. Therefore, the slope of a line parallel to it is also 4.

b) Perpendicular Line Slope: The slope of a line perpendicular to the given line is the negative reciprocal of its slope. The negative reciprocal of 4 is -1/4.

Problem 2: Determining if Lines are Parallel or Perpendicular

Question: Determine whether the lines y = 2x + 5 and y = -1/2x + 1 are parallel, perpendicular, or neither.

Solution:

The slope of the first line (y = 2x + 5) is 2. The slope of the second line (y = -1/2x + 1) is -1/2.

Since the slopes are negative reciprocals of each other (2 and -1/2), the lines are perpendicular.

Problem 3: Writing the Equation of a Parallel Line

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

Solution:

Since the line is parallel to y = 3x + 2, it has the same slope, which is 3. We can use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point.

Substituting the slope (m = 3) and the point (1, 5):

y - 5 = 3(x - 1) y - 5 = 3x - 3 y = 3x + 2

The equation of the parallel line is y = 3x + 2. Note that this is the same as the original line; this simply means that the point (1,5) already lies on the given line.

Problem 4: Writing the Equation of a Perpendicular Line

Question: Write the equation of the line that is perpendicular to y = -2x + 4 and passes through the point (2, 3).

Solution:

The slope of the given line is -2. The slope of a perpendicular line is the negative reciprocal, which is 1/2. Using the point-slope form again:

y - 3 = (1/2)(x - 2) y - 3 = (1/2)x - 1 y = (1/2)x + 2

The equation of the perpendicular line is y = (1/2)x + 2.

Problem 5: Parallel and Perpendicular Lines in Different Forms

Question: Determine if the lines 2x + 3y = 6 and 3x - 2y = 12 are parallel, perpendicular, or neither.

Solution:

First, rewrite the equations in slope-intercept form (y = mx + b):

For 2x + 3y = 6: 3y = -2x + 6 y = (-2/3)x + 2 (Slope = -2/3)

For 3x - 2y = 12: -2y = -3x + 12 y = (3/2)x - 6 (Slope = 3/2)

The slopes are negative reciprocals of each other (-2/3 and 3/2). Therefore, the lines are perpendicular.

Problem 6: Real-world Application

Question: Two streets, Main Street and Oak Street, are parallel. Main Street is represented by the equation y = 2x + 1. A new road, Elm Street, is being built perpendicular to Main Street and passes through the point (4, 6). Find the equation of the line representing Elm Street.

Solution:

Main Street has a slope of 2. Elm Street, being perpendicular, has a slope of -1/2. Using the point-slope form with the point (4, 6):

y - 6 = (-1/2)(x - 4) y - 6 = (-1/2)x + 2 y = (-1/2)x + 8

The equation of the line representing Elm Street is y = (-1/2)x + 8.

Problem 7: Advanced Problem - Determining Intersection Points

Question: Find the point of intersection of the lines y = 3x - 1 and y = -x + 5.

Solution:

Since both equations are solved for y, we can set them equal to each other:

3x - 1 = -x + 5 4x = 6 x = 3/2

Now substitute x = 3/2 into either equation to find y:

y = 3(3/2) - 1 = 9/2 - 1 = 7/2

The point of intersection is (3/2, 7/2).

Further Practice and Resources

Understanding parallel and perpendicular lines is crucial for higher-level mathematics. After completing Homework 6, consider these additional activities:

• Review: Revisit the definitions and examples provided in this guide to reinforce your understanding.
• Practice Problems: Seek out additional practice problems in your textbook or online resources.
• Visual Aids: Utilize online graphing tools to visualize parallel and perpendicular lines and their relationships. Seeing the lines graphically can greatly enhance comprehension.
• Real-World Connections: Look for examples of parallel and perpendicular lines in your everyday environment – buildings, streets, etc. This helps connect abstract concepts to the real world.

This comprehensive guide provided detailed solutions and explanations for a range of problems related to parallel and perpendicular lines. Remember, consistent practice and a solid understanding of the underlying principles are key to mastering this essential mathematical concept. Good luck!