(0, 2) y = x 1 2. y = x (2, 2) y = 2x + 2

Transcription

1 .5 Equations of Parallel and Perpendicular Lines COMMON CORE Learning Standards HSG-GPE.B.5 HSG-GPE.B. Essential Question How can ou write an equation of a line that is parallel or perpendicular to a given line and passes through a given point? Writing Equations of Parallel and Perpendicular Lines Work with a partner. Write an equation of the line that is parallel or perpendicular to the given line and passes through the given point. Use a graphing calculator to verif our answer. What is the relationship between the slopes? a. b. (0, ) (0, ) = = c. d. = + = + (, ) (, ) e. f. = + = + (0, ) (, 0) Writing Equations of Parallel and Perpendicular Lines Work with a partner. Write the equations of the parallel or perpendicular lines. Use a graphing calculator to verif our answers. a. b. MODELING WITH MATHEMATICS To be proficient in math, ou need to analze relationships mathematicall to draw conclusions. Communicate Your Answer. How can ou write an equation of a line that is parallel or perpendicular to a given line and passes through a given point?. Write an equation of the line that is (a) parallel and (b) perpendicular to the line = + and passes through the point (, ). Section.5 Equations of Parallel and Perpendicular Lines 55

2 .5 Lesson What You Will Learn Core Vocabular directed line segment, p. 5 Previous slope slope-intercept form -intercept Use slope to partition directed line segments. Identif parallel and perpendicular lines. Write equations of parallel and perpendicular lines. Use slope to find the distance from a point to a line. Partitioning a Directed Line Segment A directed line segment AB is a segment that represents moving from point A to point B. The following eample shows how to use slope to find a point on a directed line segment that partitions the segment in a given ratio. Partitioning a Directed Line Segment REMEMBER Recall that the slope of a line or line segment through two points, (, ) and (, ), is defined as follows. m = change in = change in = rise run You can choose either of the two points to be (, ). Find the coordinates of point P along the directed line segment AB so that the ratio of AP to PB is to. In order to divide the segment in the ratio to, think of dividing, or partitioning, the segment into +, or 5 congruent pieces. Point P is the point that is of the wa from point A to point B. 5 Find the rise and run from point A to point B. Leave the slope in terms of rise and run and do not simplif. slope of AB : m = 8 = = rise run To find the coordinates of point P, add of the run 5 B(, 8) to the -coordinate of A, and add 8 of the rise to the 5 -coordinate of A. run: 5 of = P(.8, 5.) 5 =.8. rise: 5 of = 5 =. A(, ).8 So, the coordinates of P are ( +.8, +.) = (.8, 5.). The ratio of AP to PB is to. 8 A(, ) B(, 8) 8 8 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio.. A(l, ), B(8, ); to. A(, ), B(, 5); to 7 5 Chapter Parallel and Perpendicular Lines

3 Identifing Parallel and Perpendicular Lines In the coordinate plane, the -ais and the -ais are perpendicular. Horizontal lines are parallel to the -ais, and vertical lines are parallel to the -ais. Theorems Theorem. Slopes of Parallel Lines In a coordinate plane, two nonvertical lines are parallel if and onl if the have the same slope. An two vertical lines are parallel. READING If the product of two numbers is, then the numbers are called negative reciprocals. Proof p. 9; E., p. Theorem. Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and onl if the product of their slopes is. m = m Horizontal lines are perpendicular to vertical lines. Proof p. 0; E., p. m m = Identifing Parallel and Perpendicular Lines Determine which of the lines are parallel and which of the lines are perpendicular. Find the slope of each line. Line a: m = 0 ( ) = Line b: m = 0 ( ) = 0 ( 5) Line c: m = = ( ) 0 Line d: m = ( ) = d (, ) (, 0) a (0, ) b (, 0) (0, ) c (, 5) (, ) Because lines b and c have the same slope, lines b and c are parallel. Because ( ) =, lines b and d are perpendicular and lines c and d are perpendicular. Monitoring Progress. Determine which of the lines are parallel and which of the lines are perpendicular. Help in English and Spanish at BigIdeasMath.com (0, ) (, 0) a b c (, ) (, ) (, ) (0, ) (, ) d Section.5 Equations of Parallel and Perpendicular Lines 57

4 Writing Equations of Parallel and Perpendicular Lines You can appl the Slopes of Parallel Lines Theorem and the Slopes of Perpendicular Lines Theorem to write equations of parallel and perpendicular lines. Writing an Equation of a Parallel Line REMEMBER The linear equation = is written in slope-intercept form = m + b, where m is the slope and b is the -intercept. Write an equation of the line passing through the point (, ) that is parallel to the line =. Step Find the slope m of the parallel line. The line = has a slope of. B the Slopes of Parallel Lines Theorem, a line parallel to this line also has a slope of. So, m =. Step Find the -intercept b b using m = and (, ) = (, ). = m + b Use slope-intercept form. = ( ) + b Substitute for m,, and. = b Solve for b. Check = + (, ) Because m = and b =, an equation of the line is = +. Use a graph to check that the line = is parallel to the line = +. Writing an Equation of a Perpendicular Line = Write an equation of the line passing through the point (, ) that is perpendicular to the line + =. Step Find the slope m of the perpendicular line. The line + =, or = +, has a slope of. Use the Slopes of Perpendicular Lines Theorem. m = The product of the slopes of lines is. m = Divide each side b. Check = + = + (, ) Step Find the -intercept b b using m = and (, ) = (, ). = m + b Use slope-intercept form. = () + b Substitute for m,, and. = b Solve for b. Because m = and b =, an equation of the line is = +. Check that the lines are perpendicular b graphing their equations and using a protractor to measure one of the angles formed b their intersection. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Write an equation of the line that passes through the point (, 5) and is (a) parallel to the line = 5 and (b) perpendicular to the line = How do ou know that the lines = and = are perpendicular? 58 Chapter Parallel and Perpendicular Lines

5 Finding the Distance from a Point to a Line Recall that the distance from a point to a line is the length of the perpendicular segment from the point to the line. Finding the Distance from a Point to a Line REMEMBER = + (, 0) Recall that the solution of a sstem of two linear equations in two variables gives the coordinates of the point of intersection of the graphs of the equations. There are two special cases when the lines have the same slope. When the sstem has no solution, the lines are parallel. When the sstem has infinitel man solutions, the lines coincide. Find the distance from the point (, 0) to the line = +. Step Find the equation of the line perpendicular to the line = + that passes through the point (, 0). First, find the slope m of the perpendicular line. The line = + has a slope of. Use the Slopes of Perpendicular Lines Theorem. m = The product of the slopes of lines is. m = Divide each side b. Then find the -intercept b b using m = and (, ) = (, 0). = m + b Use slope-intercept form. 0 = () + b Substitute for,, and m. = b Solve for b. Because m = and b =, an equation of the line is =. Step Use the two equations to write and solve a sstem of equations to find the point where the two lines intersect. = + Equation = Equation Substitute + for in Equation. = Equation + = Substitute + for. = Solve for. Substitute for in Equation and solve for. = + Equation = + Substitute for. = Simplif. So, the perpendicular lines intersect at (, ). (, ) (, 0) = + Step Use the Distance Formula to find the distance from (, 0) to (, ). distance = ( ) + (0 ) = ( ) + ( ) =. = So, the distance from the point (, 0) to the line = + is about. units. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Find the distance from the point (, ) to the line = Find the distance from the point (, ) to the line =. Section.5 Equations of Parallel and Perpendicular Lines 59

6 .5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE A line segment AB is a segment that represents moving from point A to point B.. WRITING How are the slopes of perpendicular lines related? Monitoring Progress and Modeling with Mathematics In Eercises, find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. (See Eample.). A(8, 0), B(, ); to. A(, ), B(, ); to 5. A(, ), B(, ); 5 to. A(, ), B(5, ); to In Eercises 7 and 8, determine which of the lines are parallel and which of the lines are perpendicular. (See Eample.) (, ) d (, ) a (5, ) (, ) b (, ) c (, ) (, ) (, 0) d (0, ) (, ) (, ) (, ) (, 0) a (, 0) (, ) b (0, ) In Eercises 9, tell whether the lines through the given points are parallel, perpendicular, or neither. Justif our answer. 9. Line : (, 0), (7, ) Line : (7, 0), (, ) c 0. Line : (, ), ( 7, ) Line : (, ), (8, ). Line : ( 9, ), ( 5, 7) Line : (, ), ( 7, ). Line : (0, 5), ( 8, 9) Line : (, ), (, ) In Eercises, write an equation of the line passing through point P that is parallel to the given line. Graph the equations of the lines to check that the are parallel. (See Eample.). P(0, ), = +. P(, 8), = ( + ) 5 5. P(, ), = 5. P(, 0), + = In Eercises 7 0, write an equation of the line passing through point P that is perpendicular to the given line. Graph the equations of the lines to check that the are perpendicular. (See Eample.) 7. P(0, 0), = 9 8. P(, ), = 9. P(, ), = ( + ) 0. P( 8, 0), 5 = In Eercises, find the distance from point A to the given line. (See Eample 5.). A(, 7), =. A( 9, ), =. A(5, ), 5 + =. A (, 5 ), + = 0 Chapter Parallel and Perpendicular Lines

7 5. ERROR ANALYSIS Describe and correct the error in determining whether the lines are parallel, perpendicular, or neither. Line : (, 5), (, ) Line : (0, ), (,7) ( 5) m = = m = 7 0 = Lines and are perpendicular.. ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through the point (, ) and is parallel to the line = +. = +, (, ) = m() + = m The line = + is parallel to the line = +. In Eercises 7 0, find the midpoint of PQ. Then write an equation of the line that passes through the midpoint and is perpendicular to PQ. This line is called the perpendicular bisector. 7. P(, ), Q(, ) 8. P( 5, 5), Q(, ). REASONING A triangle has vertices L(0, ), M(5, 8), and N(, ). Is the triangle a right triangle? Eplain our reasoning.. MODELING WITH MATHEMATICS A new road is being constructed parallel to the train tracks through point V. An equation of the line representing the train tracks is =. Find an equation of the line representing the new road. V 5. MODELING WITH MATHEMATICS A bike path is being constructed perpendicular to Washington Boulevard through point P(, ). An equation of the line representing Washington Boulevard is =. Find an equation of the line representing the bike path. 9. P(0, ), Q(, ) 0. P( 7, 0), Q(, 8). MODELING WITH MATHEMATICS Your school lies directl between our house and the movie theater. The distance from our house to the school is one-fourth of the distance from the school to the movie theater. What point on the graph represents our school? (5, ). PROBLEM SOLVING A gazebo is being built near a nature trail. An equation of the line representing the nature trail is =. Each unit in the coordinate plane corresponds to 0 feet. Approimatel how far is the gazebo from the nature trail? (, ) gazebo (, ) 8. REASONING Is quadrilateral QRST a parallelogram? Eplain our reasoning. Q(, ) R(, ) T(, ) S(5, ) 7. CRITICAL THINKING The slope of line is greater than 0 and less than. Write an inequalit for the slope of a line perpendicular to. Eplain our reasoning. Section.5 Equations of Parallel and Perpendicular Lines

8 8. HOW DO YOU SEE IT? Determine whether quadrilateral JKLM is a square. Eplain our reasoning. K(0, n) L(n, n) MATHEMATICAL CONNECTIONS In Eercises and, find a value for k based on the given description.. The line through (, k) and ( 7, ) is parallel to the line = +.. The line through (k, ) and (7, 0) is perpendicular to the line = 8 5. J(0, 0) M(n, 0) 5. ABSTRACT REASONING Make a conjecture about how to find the coordinates of a point that lies beond point B along AB. Use an eample to support our conjecture. 9. CRITICAL THINKING Suppose point P divides the directed line segment XY so that the ratio of XP to PY is to 5. Describe the point that divides the directed line segment YX so that the ratio of YP to PX is 5 to. 0. MAKING AN ARGUMENT Your classmate claims that no two nonvertical parallel lines can have the same -intercept. Is our classmate correct? Eplain.. MATHEMATICAL CONNECTIONS Solve each sstem of equations algebraicall. Make a conjecture about what the solution(s) can tell ou about whether the lines intersect, are parallel, or are the same line. a. = + 9 = b. + = = 8 c. = =. THOUGHT PROVOKING Find a formula for the distance from the point ( 0, 0 ) to the line a + b = 0. Verif our formula using a point and a line.. PROBLEM SOLVING What is the distance between the lines = and = + 5? Verif our answer. PROVING A THEOREM In Eercises 7 and 8, use the slopes of lines to write a paragraph proof of the theorem. 7. Lines Perpendicular to a Transversal Theorem (Theorem.): In a plane, if two lines are perpendicular to the same line, then the are parallel to each other. 8. Transitive Propert of Parallel Lines Theorem (Theorem.9): If two lines are parallel to the same line, then the are parallel to each other. 9. PROOF Prove the statement: If two lines are vertical, then the are parallel. 50. PROOF Prove the statement: If two lines are horizontal, then the are parallel. 5. PROOF Prove that horizontal lines are perpendicular to vertical lines. Maintaining Mathematical Proficienc Plot the point in a coordinate plane. (Skills Review Handbook) 5. A(, ) 5. B(0, ) 5. C(5, 0) 55. D(, ) Cop and complete the table. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons = + 9 = Chapter Parallel and Perpendicular Lines