Writing Equations of Parallel and Perpendicular Lines 4.3. Essential Question How can you recognize lines that are parallel or perpendicular?

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1 . Writing Equations of Parallel and Perpendiular Lines Essential Question How an ou reognize lines that are parallel or perpendiular? Reognizing Parallel Lines Work with a partner. Write eah linear equation in slope-interept form. Then use a graphing alulator to graph the three equations in the same square viewing window. (The graph of the first equation is shown.) Whih two lines appear parallel? How an ou tell? a. + =. 5 + = + = + = + =.5 + = 5 USING TOOLS STRATEGICALLY To e profiient in math, ou need to use a graphing alulator and other availale tehnologial tools, as appropriate, to help ou eplore relationships and deepen our understanding of onepts. CONNECTIONS TO GEOMETRY = + 5 = + Reognizing Perpendiular Lines Work with a partner. Write eah linear equation in slope-interept form. Then use a graphing alulator to graph the three equations in the same square viewing window. (The graph of the first equation is shown.) Whih two lines appear perpendiular? How an ou tell? a. + =. + 5 = 0 = + = =.5 = 5 You will learn more aout parallel and perpendiular lines and their properties in Chapter 0. = + = + 5 Communiate Your Answer. How an ou reognize lines that are parallel or perpendiular?. Compare the slopes of the lines in Eploration. How an ou use slope to determine whether two lines are parallel? Eplain our reasoning. 5. Compare the slopes of the lines in Eploration. How an ou use slope to determine whether two lines are perpendiular? Eplain our reasoning. Setion. Writing Equations of Parallel and Perpendiular Lines 77

2 . Lesson What You Will Learn Core Voaular parallel lines, p. 78 perpendiular lines, p. 7 Previous reiproal READING The phrase A if and onl if B is a wa of writing two onditional statements at one. It means that if A is true, then B is true. It also means that if B is true, then A is true. You will learn more aout these tpes of statements in Chapter. Identif and write equations of parallel lines. Identif and write equations of perpendiular lines. Use parallel and perpendiular lines in real-life prolems. Identifing and Writing Equations of Parallel Lines Core Conept Parallel Lines and Slopes Two lines in the same plane that never interset are parallel lines. Two distint nonvertial lines are parallel if and onl if the have the same slope. All vertial lines are parallel. Identifing Parallel Lines Determine whih of the lines are parallel. Find the slope of eah line. Line a: m = ( ) = 5 Line : m = 0 ( ) = 5 ( ) Line : m = ( ) = 5 Lines a and have the same slope, so the are parallel. Writing an Equation of a Parallel Line a (, ) (, 0), ) (, ) (, 5) (, ) ANOTHER WAY You an also use the slope m = and the point-slope form to write an equation of the line that passes through (5, ). = m( ) ( ) = ( 5) = Write an equation of the line that passes through (5, ) and is parallel to the line = +. Step Find the slope of the parallel line. The graph of the given equation has a slope of. So, the parallel line that passes through (5, ) also has a slope of. Step Use the slope-interept form to find the -interept of the parallel line. = m + Write the slope-interept form. = (5) + Sustitute for m, 5 for, and for. = Solve for. Using m = and =, an equation of the parallel line is =. Monitoring Progress Help in English and Spanish at BigIdeasMath.om. Line a passes through ( 5, ) and (, ). Line passes through (, ) and (, 7). Are the lines parallel? Eplain.. Write an equation of the line that passes through (, ) and is parallel to the line = Chapter Writing Linear Funtions

3 REMEMBER The produt of a nonzero numer m and its negative reiproal is : m ( m ) =. Identifing and Writing Equations of Perpendiular Lines Core Conept Perpendiular Lines and Slopes Two lines in the same plane that interset to form right angles are perpendiular lines. Nonvertial lines are perpendiular if and onl if their slopes are negative reiproals. Vertial lines are perpendiular to horizontal lines. = + = Identifing Parallel and Perpendiular Lines Determine whih of the lines, if an, are parallel or perpendiular. Line a: = + Line : + = Line : 8 = Write the equations in slope-interept form. Then ompare the slopes. Line a: = + Line : = + Line : = Lines and have slopes of, so the are parallel. Line a has a slope of, the negative reiproal of, so it is perpendiular to lines and. Writing an Equation of a Perpendiular Line ANOTHER WAY You an also use the slope m = and the slope-interept form to write an equation of the line that passes through (, ). = m + = ( ) + 5 = So, = 5. Write an equation of the line that passes through (, ) and is perpendiular to the line = +. Step Find the slope of the perpendiular line. The graph of the given equation has a slope of. Beause the slopes of perpendiular lines are negative reiproals, the slope of the perpendiular line that passes through (, ) is. Step Use the slope m = and the point-slope form to write an equation of the perpendiular line that passes through (, ). = m( ) Write the point-slope form. = [ ( )] Sustitute for m, for, and for. = = 5 Simplif. Write in slope-interept form. An equation of the perpendiular line is = 5. Monitoring Progress Help in English and Spanish at BigIdeasMath.om. Determine whih of the lines, if an, are parallel or perpendiular. Eplain. Line a: + = Line : = 8 Line : + 8 =. Write an equation of the line that passes through (, 5) and is perpendiular to the line =. Setion. Writing Equations of Parallel and Perpendiular Lines 7

4 Writing Equations for Real-Life Prolems Writing an Equation of a Perpendiular Line The position of a heliopter searh and resue rew is shown in the graph. The shortest flight path to the shoreline is one that is perpendiular to the shoreline. Write an equation that represents this path. water (, ) 8 0 shore CONNECTIONS TO GEOMETRY You will learn aout finding the distane from a point to a line in Chapter 0.. Understand the Prolem You an see the line that represents the shoreline. You know the oordinates of the heliopter. You are asked to write an equation that represents the shortest flight path to the shoreline.. Make a Plan Find the slope of the line that represents the shoreline. Use the negative reiproal of this slope, the oordinates of the heliopter, and the point-slope form to write an equation.. Solve the Prolem Step Find the slope of the line that represents the shoreline. The line passes through points (, ) and (, ). So, the slope is m = =. Beause the shoreline and shortest flight path are perpendiular, the slopes of their respetive graphs are negative reiproals. So, the slope of the graph of the shortest flight path is. Step Use the slope m = and the point-slope form to write an equation of the shortest flight path that passes through (, ). = m( ) Write the point-slope form. = ( ) Sustitute for m, for, and for. = Distriutive Propert = 7 Write in slope-interept form. An equation that represents the shortest flight path is = 7.. Look Bak To hek that our equation is orret, verif that (, ) is a solution of the equation. = () 7 Monitoring Progress Help in English and Spanish at BigIdeasMath.om 5. In Eample 5, a oat is traveling parallel to the shoreline and passes through (, ). Write an equation that represents the path of the oat. 80 Chapter Writing Linear Funtions

5 . Eerises Dnami Solutions availale at BigIdeasMath.om Voaular and Core Conept Chek. COMPLETE THE SENTENCE Two distint nonvertial lines that have the same slope are.. VOCABULARY Two lines are perpendiular. The slope of one line is 5 7. What is the slope of the other line? Justif our answer. Monitoring Progress and Modeling with Mathematis In Eerises 8, determine whih of the lines, if an, are parallel. Eplain. (See Eample.).. (, ) (, ) (0, ) a (0, 0) (, 0) (, ) 5. Line a passes through (, ) and (, 0). Line passes through (, ) and (, ). Line passes through (0, ) and (, ). (, 0) a (5, 0). Line a passes through (, ) and (, ). Line passes through (, ) and (, ). Line passes through (, 8) and (, 0). (0, 5) (, ) (, ) (5, ) 5. Line a passes through (, ) and (0, ). Line passes through (, ) and (, ). Line passes through (, ) and (, ).. Line a passes through (, 0) and (, ). Line passes through (, ) and (, ). Line passes through (, 0) and (, ). 7. Line a: = Line : = + Line : + = 8. Line a: = Line : = Line : + = In Eerises, write an equation of the line that passes through the given point and is perpendiular to the given line. (See Eample.). (7, 0); = 0. (, ); = +. (, ); = 8. (8, ); + = 7. Line a: + = 8 Line : + = Line : = + 8. Line a: = Line : = + 8 Line : = In Eerises and, write an equation of the line that passes through the given point and is (a) parallel and () perpendiular to the given line. In Eerises, write an equation of the line that passes through the given point and is parallel to the given line. (See Eample.). (, ); = + 0. (, ); = 5 +. (8, ); =. (, 5); = + 0 In Eerises 8, determine whih of the lines, if an, are parallel or perpendiular. Eplain. (See Eample.).. (, ) (0, ) (, ) ( 5, ) (, ) (, ) a (0, 5) (, ) (, 5) (, ) (, 0) 5 (0, 0) a.. 5. ERROR ANALYSIS Desrie and orret the error in writing an equation of the line that passes through (, ) and is parallel to the line = +. (, ) = m( ) = ( ) = + = (, ) Setion. Writing Equations of Parallel and Perpendiular Lines 8

6 . ERROR ANALYSIS Desrie and orret the error in writing an equation of the line that passes through (, 5) and is perpendiular to the line = + 5. = m( ) ( 5) = ( ) + 5 = = 7. MAKING AN ARGUMENT A hoke puk leaves the lade of a hoke stik, ounes off a wall, and travels in a new diretion, as shown. Your friend laims the path of the puk forms a right angle. Is our friend orret? Eplain. (, ) (0, 8) 8 (, 0) 7. MODELING WITH MATHEMATICS A it water department is proposing the onstrution of a new water pipe, as shown. The new pipe will e perpendiular to the old pipe. Write an equation that represents the new pipe. (See Eample 5.) Softall Aadem (, 0) 8 proposed water pipe Total ost (dollars) eisting (0, ) water pipe onnetor valve (, ) students an initial registration fee plus a monthl fee. The graph shows the total amounts paid two students over a -month period. The lines are parallel.. HOW DO YOU SEE IT? A softall aadem harges Student B Student A 0 Months of memership 8. MODELING WITH MATHEMATICS A parks and rereation department is onstruting a new ike path. The path will e parallel to the railroad traks shown and pass through the parking area at the point (, 5). Write an equation that represents the path. 8 a. Did one of the students pa a greater registration fee? Eplain.. Did one of the students pa a greater monthl fee? Eplain. parking area (, 5) (, ) REASONING In Eerises 5, determine whether the statement is alwas, sometimes, or never true. Eplain our reasoning.. Two lines with positive slopes are perpendiular. (8, 0) 0. MATHEMATICAL CONNECTIONS The verties of a quadrilateral are A(, ), B(, ), C(8, 0), and D(, 8). a. Is quadrilateral ABCD a parallelogram? Eplain.. Is quadrilateral ABCD a retangle? Eplain. 0. USING STRUCTURE For what value of a are the graphs of = + and = a 5 parallel? perpendiular? Maintaining Mathematial Profiien. A vertial line is parallel to the -ais. 5. Two lines with the same -interept are perpendiular.. THOUGHT PROVOKING You are designing a new logo for our math lu. Your teaher asks ou to inlude at least one pair of parallel lines and at least one pair of perpendiular lines. Sketh our logo in a oordinate plane. Write the equations of the parallel and perpendiular lines. Reviewing what ou learned in previous grades and lessons Determine whether the relation is a funtion. Eplain. (Setion.) 7. (, ), (, 8), (5, 0), (, ), (7, ) 8. (, ), (, ), (, ), (, ), (, 5) 8 Chapter int_math_pe_00.indd 8 Writing Linear Funtions //5 : PM