4.5 Equations of Parallel and Perpendicular Lines

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1 Name Class Date.5 Equations of Parallel and Perpendicular Lines Essential Question: How can ou find the equation of a line that is parallel or perpendicular to a given line? Resource Locker Eplore Eploring Slopes of Lines Recall that the slope of a straight line in a coordinate plane is the _ ratio of the rise to the run. In the figure, the slope of rise AB is run = 8 = _ A(-, -3) B(, 1) 3 Rise = 1-(-3) = -5 Run = -(-) = 8 A Graph the equations = ( + 1) and = - 3. B What do ou notice about the graphs of the two lines? About the slopes of the lines? Houghton Mifflin Harcourt Publishing Compan C D The graphs of + 3 = and = 3-1 are shown. Use a protractor. What is the measure of the angle formed b the intersection of the lines. What does that tell ou about the lines? What are the slopes of the two lines? How are the related? E Complete the statements: If two nonvertical lines are, then the have equal slopes. If two nonvertical lines are perpendicular, then the product of their slopes is. Module 05 Lesson 5

2 Reflect 1. Your friend sas that if two lines have opposite slopes, the are perpendicular. He uses the slopes 1 and 1 as eamples. Do ou agree with our friend? Eplain.. The frets on a guitar are all perpendicular to one of the strings. Eplain wh the frets must be parallel to each other. Eplain 1 Writing Equations of Parallel Lines You can use slope relationships to write an equation of a line parallel to a given line. Eample 1 Write the equation of each line in slope-intercept form. A The line parallel to = that passes through (-1, ) Parallel lines have equal slopes. So the slope of the required line is 5. Use point-slope form. - 1 = m ( - 1 ) Substitute for m, 1, 1. - = 5 ( - (-1)) Simplif. - = Solve for. = The equation of the line is = B The line parallel to = -3 + that passes through (9, -6) Parallel lines have slopes. So the slope of the required line is. Use point-slope form. - 1 = m( - 1 ) Substitute for m, 1, 1. - = ( - ) Simplif. + 6 = + Solve for. = + The equation of the line is. Houghton Mifflin Harcourt Publishing Compan Module 06 Lesson 5

3 Reflect 3. What is the equation of the line through a given point and parallel to the -ais? Wh? Your Turn Write the equation of each line in slope-intercept form.. The line parallel to = - that passes through (5,.5) 5. The line parallel to = 3 + tha t passes through (-, 0) Eplain Writing Equations of Perpendicular Lines You can use slope relationships to write an equation of a line perpendicular to a given line. Eample Write the equation of each line in slope-intercept form. Houghton Mifflin Harcourt Publishing Compan A The line perpendicular to = - that passes through (3, -1) Perpendicular lines have slopes that are opposite reciprocals, which means that the product of the slopes will be -1. So the slope of the required line is - _. - 1 = m ( - 1 ) Use point-slope form. - (-1) = - ( - 3) Substitute for m, 1, = - + 3_ Simplif. = - - Solve for. The equation of the line is = - _ - _. B The line perpendicular to = that passes through (-6, -8) 5 The product of the slopes of perpendicular lines is. So the slope of the required line is. - 1 = m ( - 1 ) Use point-slope form. - = ( - ) Substitute for m, 1, = + Simplif. = + Solve for. The equation of the line is. Module 07 Lesson 5

Reflect 6. A carpenter s square forms a right angle. A carpenter places the square so that one side is parallel to an edge of a board, and then draws a line along the other side of the square.

4 Reflect 6. A carpenter s square forms a right angle. A carpenter places the square so that one side is parallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the square to the right and draws a second line. Wh must the two lines be parallel? Your Turn Write the equation of each line in slope-intercept form. 7. The line perpendicular to = 3 + that 8. The line perpendicular to = - that passes through (3, 1) passes through (0, 0) Elaborate 9. Discussion Would it make sense to find the equation of a line parallel to a given line, and through a point on the given line? Eplain. 10. Would it make sense to find the equation of a line perpendicular to a given line, and through a point on the given line? Eplain. 11. Essential Question Check-In How are the slopes of parallel lines and perpendicular lines related? Assume the lines are not vertical. Houghton Mifflin Harcourt Publishing Compan Image Credits: Zoran Zeremski/Shutterstock Module 08 Lesson 5

Evaluate: Homework and Practice Use the graph for Eercises 1. 1. A line with a positive slope is parallel to one of the lines shown. What is its slope?

5 Evaluate: Homework and Practice Use the graph for Eercises A line with a positive slope is parallel to one of the lines shown. What is its slope?. A line with a negative slope is perpendicular to one of the lines shown. What is its slope? 6 Online Homework Hints and Help Etra Practice 3. A line with a positive slope is perpendicular to one of the lines shown. What is its slope? 0 6. A line with a negative slope is parallel to one of the lines shown. What is its slope? Find the equation of the line that is parallel to the given line and passes through the given point. 5. = 3 + 1; (9, 0) 6. = 0.6 3; (, ) 7. = 5 ( + 1) ; (, - ) Houghton Mifflin Harcourt Publishing Compan Find the equation of the line that is perpendicular to the given line and passes through the given point. 8. = 10; (1, -3) 9. = - 3-5; (1, 0) 10. = _ ; (1, 1) 3 Module 09 Lesson 5

11. Determine whether the lines are parallel. Use slope to eplain our answer. - - 0 - - The endpoints of a side of rectangle ABCD in the coordinate plane are at A (1, 5) and B (3, 1).

6 11. Determine whether the lines are parallel. Use slope to eplain our answer The endpoints of a side of rectangle ABCD in the coordinate plane are at A (1, 5) and B (3, 1). Find the equation of the line that contains the given segment. 1. _ AB 13. _ BC 1. _ AD 15. _ CD if point C is at (7, 3) 16. A well is to be dug at the location shown in the diagram. Use the diagram for parts (a c). a. Find the equation that represents the road. Well b. A path is to be made from the road to the well. Describe how this should be done to minimize the length of the path. c. Find the equation of the line that contains the path. - - Road Houghton Mifflin Harcourt Publishing Compan Image Credits: Gar S. Chapman/Photographer's Choice RF/Gett Images Module 10 Lesson 5

7 17. Use the graph for parts (a c), a. Find the equation of the perpendicular bisector of the segment. Eplain our method b. Find the equation of the line that is parallel to the segment, but has the same -intercept as the equation ou found in part a. c. What is the relationship between the two lines ou found in parts (a) and (b)? Line m is perpendicular to - 3 = -1 and passes through (1, 5). What is the slope of line m? A. -3 B. C. 3 D Determine whether each pair of lines are parallel, perpendicular, or neither. Select the correct answer for each lettered part. a. - = 1; = + 5 Parallel Perpendicular Neither b. + = 8; = 5 Parallel Perpendicular Neither 5 c. 3 - = 1; 3 = Parallel Perpendicular Neither d. = 3-1; 15-5 = 10 Parallel Perpendicular Neither e. 7 = + 1; = 10 Parallel Perpendicular Neither H.O.T. Focus on Higher Order Thinking Houghton Mifflin Harcourt Publishing Compan 0. Communicate Mathematical Ideas Two lines in the coordinate plane have opposite slopes, are parallel, and the sum of their -intercepts is 10. If one of the lines passes through (5, ), what are the equations of the lines? 1. Eplain the Error Alan sas that two lines in the coordinate plane are perpendicular if and onl if the slopes of the lines are m and 1. Identif and correct two errors in Alan s statement. m. Analze Relationships Two perpendicular lines have opposite -intercepts. The equation of one of these lines is = m + b. Epress the -coordinate of the intersection point of the lines in terms of m and b. Module 11 Lesson 5

8 Lesson Performance Task Surveors tpicall use a unit of measure called a rod, which equals 16 _ feet. (A rod ma seem like an odd unit, but it s ver useful for measuring sections of land, because an acre equals eactl 160 square rods.) A surveor was called upon to find the distance between a new interpretive center at a park and the park entrance. The surveor plotted the points shown on a coordinate grid of the park in units of 1 rod. The line between the Interpretive Center and Park Headquarters forms a right angle with the line connecting the Park Headquarters and Park Entrance. What is the distance, in feet, between the Interpretive Center and the park entrance? Eplain the process ou used to find the answer. W Interpretive Center N S Park Entrance (5, 5) E Park Headquarters (15, 0) Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson 5

Equations of Parallel and Perpendicular Lines

COMMON CORE AB is rise - - 1 - - 0 - - 8 6 Locker LESSON. Equations of Parallel and Perpendicular Lines Name Class Date. Equations of Parallel and Perpendicular Lines Essential Question: How can ou find