Equations of Parallel and Perpendicular Lines

Transcription

1 COMMON CORE AB is rise Locker LESSON. Equations of Parallel and Perpendicular Lines Name Class Date. 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 Common Core Math Standards The student is epected to: COMMON CORE G-GPE.B. find the equation of a line parallel or perpendicular to a given line that passes through a given point. Mathematical Practices COMMON CORE MP. Reasoning Language Objective Eplain to a partner how to use the slope of a line to find the equation of a parallel or perpendicular line. ENGAGE Essential Question: How can ou find the equation of a line that is parallel or perpendicular to a given line? Possible answer: The slopes of parallel lines are equal. Substitute the known slope and the coordinates of a point on the other line into the point-slope form to find the equation of the parallel line. The product of the slopes of perpendicular lines is -1. Substitute the opposite reciprocal of the known slope and the coordinates of a point on the other line into the point-slope form to find the equation of the perpendicular line. Houghton Mifflin Harcourt Publishing Compan 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 AB is rise run = 8 =. A B C D E Graph the equations = ( + 1) and = -. What do ou notice about the graphs of the two lines? About the slopes of the lines? The lines are parallel. The slopes are equal. The graphs of + = and = - 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? 90 ; the lines are perpendicular. What are the slopes of the two lines? How are the related? - and ; the slopes are opposite reciprocals A(-, -) Complete the statements: If two nonvertical lines are parallel, then the have equal slopes. If two nonvertical lines are perpendicular, 1 then the product of their slopes is B(, 1) Rise = 1-(-) = Run = -(-) = PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo. Eplain that GPS stands for Global Positioning Sstem, a sstem of orbiting satellites that enables a person to pinpoint his or her precise location on Earth s surface. Then preview the Lesson Performance Task. GEMNLESE879UM0L.indd 0 Module 0 Lesson DO NOT EDIT--Changes must be made through "File info" CorrectionKe=NL-D;CA-D Name Class Date. 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? G-GPE.B. find the equation of a line parallel or perpendicular to a given line that passes through a given point. Houghton Mifflin Harcourt Publishing Compan 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 run = 8 =. Graph the equations = ( + 1) and = -. What do ou notice about the graphs of the two lines? About the slopes of the lines? The lines are parallel. The slopes are equal. The graphs of + = and = - 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? 90 ; the lines are perpendicular. What are the slopes of the two lines? How are the related? - and ; the slopes are opposite reciprocals. Resource B(, 1) Rise = 1-(-) A(-, -) = Run = -(-) = 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. parallel 1 Module 0 Lesson GEMNLESE879UM0L.indd 0 //16 11:7 PM HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. //16 11:7 PM 0 Lesson.

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. No; although lines with slopes of 1 and -1 are perpendicular, it s because the product of the slopes is -1. Slopes of and - are opposites, but the corresponding lines are not perpendicular.. The frets on a guitar are all perpendicular to one of the strings. Eplain wh the frets must be parallel to each other. The frets are lines that are perpendicular to the same line (the string), so 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. The line parallel to = + 1 that passes through (-1, ) Parallel lines have equal slopes. So the slope of the required line is. Use point-slope form. - 1 = m ( - 1 ) Substitute for m, 1, 1. - = ( - (-1)) EXPLORE Eploring Slopes of Lines INTEGRATE TECHNOLOGY Students have used geometr software to construct perpendicular lines and calculate their slopes. The can use the calculation feature to find the product of slopes of perpendicular lines is alwas -1. QUESTIONING STRATEGIES What appears to be true about the slopes of non-vertical parallel lines? The are equal. What appears to be true about the slopes of two non-vertical perpendicular lines? The slopes are opposite reciprocals. Simplif. - = + Solve for. = + 7 The equation of the line is = + 7. The line parallel to = - + that passes through (9, -6) Parallel lines have equal slopes. So the slope of the required line is -. Use point-slope form. - 1 = m( - 1 ) Substitute for m, 1, = ( - ) Simplif. + 6 = Solve for. = The equation of the line is = Houghton Mifflin Harcourt Publishing Compan Focus on Critical Thinking MP. You ma want to discuss the biconditional nature of the slope criteria. Because the are if and onl if statements, the criteria can be used in either direction. That is, if ou know that two lines are parallel (perpendicular), ou can conclude that the have the same (opposite reciprocal) slope. Conversel, if ou know that two lines have the same (opposite reciprocal) slope, ou can conclude that the are parallel (perpendicular). Module 06 Lesson PROFESSIONAL DEVELOPMENT Math Background In this lesson, students use the slope criterion for parallel lines and the slope criterion for perpendicular lines to solve problems. Note that the slope criteria given here assume that the lines are neither vertical nor horizontal. If the lines are vertical, the criteria for parallel and perpendicular lines do not appl, since slope is not defined for vertical lines. If the lines are horizontal, both lines have a slope of zero, and the criterion for parallel lines is trivial. EXPLAIN 1 Writing Equations of Parallel Lines QUESTIONING STRATEGIES How do ou know if an equation is written in slope-intercept form? It is of the form = m + b, with m the slope and b the -intercept. How can ou use graphing to check our answer? Graph the given line and our answer line. The should be parallel. Equations of Parallel and Perpendicular Lines 06

3 AVOID COMMON ERRORS Remind students that the -coefficient gives the slope of a line onl when the equation of the line is written in slope-intercept form. For eample, some students might sa that the slope of the line represented b the equation - = is -. However, the equation is not in slope-intercept form. Rewriting the equation in this form gives = +, which shows that the correct slope is. EXPLAIN Writing Equations of Perpendicular Lines Reflect. What is the equation of the line through a given point and parallel to the -ais? Wh? The equation is = 1, where 1 is the -coordinate of the given point. This is because the -ais is a horizontal line with equation = 0. Your Turn Write the equation of each line in slope-intercept form.. The line parallel to = - that passes through (,.) -. = -1 ( - ) -. = - + Eplain. The line parallel to = + tha t passes through (-, 0) 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. - (0) = ( - (-)) = + 6 Focus on Technolog MP. Students can use their graphing calculators to check that two equations represent perpendicular lines. However, students should be aware that perpendicular lines ma or ma not appear to be perpendicular on a graphing calculator, depending upon the viewing window that is used. To ensure that perpendicular lines appear to be perpendicular, students should go to the ZOOM menu and choose :ZSquare. QUESTIONING STRATEGIES The given line has a positive slope. What does this tell ou about the required perpendicular line? Wh? It must have a negative slope because the product of the slopes is 1. How can ou check our answers? Check that the product of the slopes is 1. Houghton Mifflin Harcourt Publishing Compan The line perpendicular to = - that passes through (, -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 = m ( - 1 ) Use point-slope form. - (-1) = - ( - ) Substitute for m, 1, = - + Simplif. = - - Solve for. The equation of the line is = - -. The line perpendicular to = that passes through (-6, -8) The product of the slopes of perpendicular lines is -1. So the slope of the required line is. - 1 = m ( - 1 ) Use point-slope form = + 1 Simplif. = + 7 Solve for. The equation of the line is = = ( - ) Substitute for m, 1, 1. Module 07 Lesson COLLABORATIVE LEARNING Whole Class Activit Have groups of students create posters to describe the slope criteria. Then remind students about the biconditional nature of the criteria and ask them if the criteria are true in two directions. Ask them to displa the criteria as graphic organizers. Sample organizers: Lines are parallel. Lines are perpendicular. Lines have the same slope. The product of slopes is Lesson.

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? Both lines are perpendicular to the edge of the board. If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other, so the lines must be parallel to each other. Your Turn Write the equation of each line in slope-intercept form. 7. The line perpendicular to = + that 8. The line perpendicular to = - that passes through (, 1) passes through (0, 0) (-1) = - ( - ) = = ( - 0) = ELABORATE QUESTIONING STRATEGIES What is the equation of the line through a given point and parallel to the -ais? Wh? The equation is = b, where b is the -coordinate of the point. Can either of the lines referred to in the slope criterion for perpendicular lines be vertical? Wh or wh not? No; the slope criterion specifies that neither line is vertical. However, since the lines are perpendicular, if one line were horizontal, the other would be vertical. 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. No; if the point is on the line, the line can t be parallel to the line, because it either intersects it or it is the same line. 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. Yes; the line will be perpendicular to the given line at the point. 11. Essential Question Check-In How are the slopes of parallel lines and perpendicular lines related? Assume the lines are not vertical. Parallel lines have the same slope; perpendicular lines have slopes whose product is -1. Houghton Mifflin Harcourt Publishing Compan Image Credits: Zoran Zeremski/Shutterstock SUMMARIZE THE LESSON Given the equation of a line and a point not on the line, how do ou find the equation of a line parallel and perpendicular to the given line? Sample answer: Parallel: Use the slope-intercept form of a line, = m + b, and replace m with the slope from the given line. Use the given point and the slope to solve for b and then rewrite the equation using the same slope m and the new -intercept b. Perpendicular: Do the same steps as for parallel ecept replace m with the opposite reciprocal of m. Module 08 Lesson DIFFERENTIATE INSTRUCTION Communicating Math Group students in pairs and give each pair a sheet of graph paper with a non-vertical, non-horizontal line drawn on it. Have students draw aes and find the equation of the line. Then have each student plot a point that is not on the line, and find the equation of the line that is parallel, and the equation of the line that is perpendicular to the original line and that passes through the partner s point. When the are done, the should compare the slopes of their lines to show that the two new lines are parallel (or perpendicular) to each other. Equations of Parallel and Perpendicular Lines 08

5 EVALUATE Evaluate: Homework and Practice ASSIGNMENT GUIDE Concept & Skills Eplore Eploring Slopes of Lines Eample 1 Writing Equations of Parallel Lines Eample Writing Equations of Perpendicular Lines Practice Eercises 1 Eercises 7, 11 1, 1 1 Eercises 8 10, 1, 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? -. A line with a positive slope is perpendicular to one of the lines shown. What is its slope? The line will be perpendicular to the line with slope -1 = -. So the slope is 1-.. A line with a negative slope is parallel to one of the lines shown. What is its slope? The line will be parallel to the line with slope -1 = -. So the slope is Online Homework Hints and Help Etra Practice 6 AVOID COMMON ERRORS A common error students make when finding slopes of perpendicular lines is using the same sign for both slopes. One slope must be the opposite reciprocal of the other, not just the reciprocal of the other, so that the product is -1, not 1. Houghton Mifflin Harcourt Publishing Compan Find the equation of the line that is parallel to the given line and passes through the given point.. = + 1; (9, 0) 6. = 0.6 ; (, ) 7. = ( + 1) ; (, - 0 = ( 9) = + 7 = 0.6 ( ( ) ) = ) = + ( ) = ( ) = Find the equation of the line that is perpendicular to the given line and passes through the given point. 8. = 10; (1, -) 9. = - - ; (1, 0) 10. = + 1 ; (1, 1) - (-) = -0.10( - 1) = ( - 1) = + 1 = = - 6 = = - ( - 1) = Module 09 Lesson Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1 1 Recall of Information MP.6 Precision 10 1 Recall of Information MP. Reasoning 11 Skills/Concepts MP. Reasoning 1 1 Skills/Concepts MP. Modeling 16 Skills/Concepts MP. Modeling Skills/Concepts MP. Reasoning 09 Lesson.

6 11. Determine whether the lines are parallel. Use slope to eplain our answer. The top line passes through (-, 0) and (0, ), so its slope is. The bottom line passes through (0, -). The lines do not have the and (, 0), so its slope is same slope, so the are not parallel. The endpoints of a side of rectangle ABCD in the coordinate plane are at A (1, ) and B (, 1). Find the equation of the line that contains the given segment. 1. AB 1. BC The slope of the required line is = - ( - ) ; - 1 = - + 6; = 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 intercept is -. - Road = - 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 BC AB, so the slope of the required line is. - 1 = ( - ) ; - 1 = - ; = - 1. AD 1. CD if point C is at (7, ) AD BC, CD so the slope of the required line is AB, so the slope of the required. line is -. - = -( - 7) ; = = ( - 1) 9 ; = + The slope is -6 - (-) and the = The line containing the path should be perpendicular to the road. - (-7) = - (-) ; = - - Module 10 Lesson Eercise Depth of Knowledge (D.O.K.) 0 Strategic Thinking MP. Reasoning COMMON CORE Mathematical Practices Houghton Mifflin Harcourt Publishing Compan Image Credits: Gar S. Chapman/Photographer's Choice RF/Gett Images Focus on Critical Thinking MP. Ask students to think about how the can use slope to solve geometr problems. Remind them that since parallel lines have the same slope, the can analze lines containing sides of polgons, for eample, to see whether the sides are parallel. The can also use slope to write the equation of a line that is parallel or perpendicular to a given line. Point out that since perpendicular lines have opposite reciprocal slopes, the can analze the lines containing the sides of polgons, for eample, to see if the polgon contains an right angles. Focus on Technolog MP. Ask each student to use geometr software to draw a simple sketch that involves perpendicular lines and right angles. Have students echange sketches, measure the slopes of the lines, and find the product of the slopes. Have them do another sketch that involves parallel lines. Have students echange sketches and measure the slopes of the lines. Focus on Math Connections MP.1 Instruct students to write two equations in the form = m + b, one with a positive value of m and one with a negative value of m. Have them graph each line on a coordinate plane and then plot a point that is not on either line. For each of the original lines, instruct students to eplain how to find the equation of a parallel line through the point and the perpendicular line through the point. 1 Skills/Concepts MP. Logic Strategic Thinking MP. Reasoning Equations of Parallel and Perpendicular Lines 10

7 COOPERATIVE LEARNING Have students work in groups of three or four. Ask them to choose one student to give directions. Instruct the other students to each draw a line and a point that is not on the line. The first student will then give a set of directions, step b step, for finding the equation of a parallel or of a perpendicular to the line through the point. The other students will not know which tpe of line it is until the have followed the directions. Once the student has successfull guided the other students through the process to find the equation of the line, have another student take the lead and describe the process to the others. JOURNAL Have students write and solve a problem involving finding the equation of a line that is parallel to a given line. Remind students to show all the steps of the solution and to eplain how the can check the answer. 17. Use the graph for parts (a c), a. Find the equation of the perpendicular bisector of the segment. 10 Eplain our method m = 0-10 ( = -1; midpt M = , ) = (7, 10) ; 90 perpendicular bisector: - 10 = 1 ( - 7) or = + 0 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. slope = -1: = c. What is the relationship between the two lines ou found in parts (a) and (b)? The are perpendicular. 18. Line m is perpendicular to - = -1 and passes through (1, ). What is the slope of line m? A. - B. C. D. A; the slope of the given line is and its opposite reciprocal is Determine whether each pair of lines are parallel, perpendicular, or neither. Select the correct answer for each lettered part. a. - = 1; = + Parallel Perpendicular Neither b. + = 8; = Parallel Perpendicular Neither c. - = 1; = - + Parallel Perpendicular Neither d. = - 1; 1 - = 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 (, ), what are the equations of the lines? = and = 6; parallel lines have equal slopes if the slopes are opposites, the must be zero; a line with slope 0 through (, ) has equation =. 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 m 1. Identif and correct two errors in Alan s statement. He should have said two nonvertical lines because vertical lines have undefined slope. He should have had a negative sign on one of his epressions for slope because the slopes of nonvertical perpendicular lines have a product of -1.. 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. If one equation is = m + b, then the other is = - m - b. m + b = - m - b m + m = -b; ( m + 1) = -mb; = -mb m + 1 Module 11 Lesson 11 Lesson.

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. About 18 feet; one method is shown. W Interpretive Center N S Park Entrance (, ) E Park Headquarters (1, 0) Find the slope of the line between Park Headquarters (PH) and the park entrance (PE): m = = 10 = Since the line connecting the Interpretive Center (IC) and Park Headquarters (PH) forms a right angle with the line connecting PH and PE, its slope must be the opposite reciprocal, -. Use the slope and the coordinates of PH (1, 0) to find the coordinates of IC: m = = = = - -6 = - = 6 So the coordinates of IC are (0, 6). Use the distance formula to find the distance in rods between IC and PE: d = ( - 0) + ( - 6) 1. So the distance in feet equals , or approimatel 18 feet. Module 1 Lesson EXTENSION ACTIVITY Houghton Mifflin Harcourt Publishing Compan AVOID COMMON ERRORS In calculating the distance between the park entrance and the Interpretive Center, students ma appl the Distance Formula incorrectl as: d = ( - 0) + ( - 6) = + 19 = + 19 = + 19 = Remind students to find the sum beneath the radical sign first before finding the square root: d = ( - 0) + ( - 6) = + 19 = = 986 Focus on Critical Thinking MP. The distance between Park Headquarters and the Interpretive Center is approimatel 67 feet. Eplain how ou could find the distance from the park entrance to Park Headquarters without using the Distance Formula. You know the distance from the Interpretive Center to Park Headquarters (67 feet) and the distance from the Interpretive Center to the park entrance (18 feet), the lengths of two sides of a right triangle. You can use the Pthagorean Theorem to find the length of the third side joining the park entrance and Park Headquarters. The Lesson Performance Task mentions that an acre equals 160 square rods. Have students show that that is true, using the fact that 1 square mile equals 60 acres. Possible eplanation: 1 mi = 80 ft so 1 m i = = 7, 878, 00 f t 1 acre = 7, 878, 00 f t 60 =, 60 f t 1 rod = 16. ft, so 1 ro d = = 7. f t 1 acre = (, 60 7.) = 160 ro d Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Equations of Parallel and Perpendicular Lines 1