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

Transcription

1 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G..B G..C Equations of Parallel and Perpendicular Lines 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. a. b. (0, ) (0, ) = x = x c. d. = x + = x + (, ) (, ) e. f. = x + = x + (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. APPLYING 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 = x + and passes through the point (, ). Section. Equations of Parallel and Perpendicular Lines

2 . Lesson What You Will Learn Core Vocabular Previous slope-intercept form -intercept Write equations of parallel and perpendicular lines. Use slope to find the distance from a point to a line. Writing Equations of Parallel and Perpendicular Lines You can appl the Slopes of Parallel Lines Theorem (Theorem.) and the Slopes of Perpendicular Lines Theorem (Theorem.) to write equations of parallel and perpendicular lines. Writing an Equation of a Parallel Line REMEMBER The linear equation = x is written in slope-intercept form = mx + 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 = x. SOLUTION Step Find the slope m of the parallel line. The line = x has a slope of. B the Slopes of Parallel Lines Theorem (Theorem.), a line parallel to this line also has a slope of. So, m =. Step Find the -intercept b b using m = and (x, ) = (, ). = mx + b Use slope-intercept form. = ( ) + b Substitute for m, x, and. = b Solve for b. Because m = and b =, an equation of the line is = x +. Use a graph to check that the line = x is parallel to the line = x +. Check = x + (, ) x = x Monitoring Progress. Write an equation of the line that passes through the point (, 5) and is parallel to the given line. Graph the equations of the lines to check that the are parallel. Help in English and Spanish at BigIdeasMath.com x = x 5 Chapter Parallel and Perpendicular Lines

3 Writing an Equation of a Perpendicular Line Write an equation of the line passing through the point (, ) that is perpendicular to the given line. x + = x SOLUTION Step Find the slope m of the perpendicular line. The line x + =, or = x +, has a slope of. Use the Slopes of Perpendicular Lines Theorem (Theorem.). m = The product of the slopes of lines is. m = Divide each side b. Step Find the -intercept b b using m = and (x, ) = (, ). = mx + b Use slope-intercept form. = () + b Substitute for m, x, and. = b Solve for b. Because m = and b =, an equation of the line is = x +. Check that the lines are perpendicular b graphing their equations and using a protractor to measure one of the angles formed b their intersection. Check = x + = x + (, ) x Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Write an equation of the line that passes through the point (, 7) and is perpendicular to the line = x + 9. Graph the equations of the lines to check that the are perpendicular.. How do ou know that the lines x = and = are perpendicular? Section. Equations of Parallel and Perpendicular Lines

4 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 = x + (, 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. x Find the distance from the point (, 0) to the line = x +. SOLUTION Step Find the equation of the line perpendicular to the line = x + that passes through the point (, 0). First, find the slope m of the perpendicular line. The line = x + has a slope of. Use the Slopes of Perpendicular Lines Theorem (Theorem.). m = The product of the slopes of lines is. m = Divide each side b. Then find the -intercept b b using m = and (x, ) = (, 0). = mx + b Use slope-intercept form. 0 = () + b Substitute for x,, and m. = b Solve for b. Because m = and b =, an equation of the line is = x. Step Use the two equations to write and solve a sstem of equations to find the point where the two lines intersect. = x + Equation = x Equation Substitute x + for in Equation. = x Equation x + = x Substitute x + for. x = Solve for x. Substitute for x in Equation and solve for. = x + Equation = + Substitute for x. = Simplif. So, the perpendicular lines intersect at (, ). (, ) x (, 0) = x + Step Use the Distance Formula to find the distance from (, 0) to (, ). distance = ( ) + (0 ) = ( ) + ( ) =. = x So, the distance from the point (, 0) to the line = x + is about. units. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Find the distance from the point (, ) to the line = x Find the distance from the point (, ) to the line = x. Chapter Parallel and Perpendicular Lines

5 . Exercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE To find the distance from point P to line g, ou must first find the equation of the line to line g that passes through point P.. WRITING Explain how to write an equation of the line that passes through the point (, ) and is (a) parallel to the line = 5 and (b) perpendicular to the line = 5. Monitoring Progress and Modeling with Mathematics In Exercises, 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 Example.). P(0, ), = x +. P(, 8), = (x + ) 5 5. P(, ), x = 5. P(, 0), x + = In Exercises 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 Example.) 7. P(0, 0), = 9x 8. P(, ), =. MODELING WITH MATHEMATICS A new road is being constructed parallel to train tracks through point V (, ). An equation of the line representing the train tracks is = x. Find an equation of the line representing the new road. 7. 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 = x. Find an equation of the line representing the bike path. 9. P(, ), = (x + ) 0. P( 8, 0), x 5 = In Exercises, find the distance from point A to the given line. (See Example.). A(, 7), = x. A( 9, ), = x. A(5, ), 5x + =. A (, 5 ), x + = 5. 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 = x +. = x +, (, ) = m() + = m The line = x + is parallel to the line = x PROBLEM SOLVING A gazebo is being built near a nature trail. An equation of the line representing the nature trail is = x. Each unit in the coordinate plane corresponds to 0 feet. Approximatel how far is the gazebo from the nature trail? gazebo (, ) 8 x Section. Equations of Parallel and Perpendicular Lines 5

6 In Exercises 9, find the midpoint of PQ. Then write. MATHEMATICAL CONNECTIONS Solve each sstem an equation of the line that passes through the midpoint Dnamic Solutions available at BigIdeasMath.com and is perpendicular to PQ of equations algebraicall. Make a conjecture about. This line is called the what the solution(s) can tell ou about whether the perpendicular bisector. lines intersect, are parallel, or are the same line. 9. P(, ), Q(, ) 0. P( 5, 5), Q(, ) a. = x + 9 x =. P(0, ), Q(, ). P( 7, 0), Q(, 8). MODELING WITH MATHEMATICS Two cars are traveling at a speed of 0 miles per hour. The second car is miles ahead of the first car. b. + x = x = 8 c. = 5x + 0x + = a. Write and graph a linear equation that represents the position p of the first car after t hours. b. Write and graph a linear equation that represents the position p of the second car after t hours. c. Are the lines in parts (a) and (b) parallel, perpendicular, or neither? Explain our reasoning.. HOW DO YOU SEE IT? Determine whether each equation is the equation of a line parallel to the given line, perpendicular to the given line, or neither. Explain our reasoning. = x + a. = x b. = x + c. = x + d. = x 5. MAKING AN ARGUMENT Your classmate claims that no two nonvertical parallel lines can have the same -intercept. Is our classmate correct? Explain. x MATHEMATICAL CONNECTIONS In Exercises 7 and 8, find a value for k based on the given description. 7. The line through (, k) and ( 7, ) is parallel to the line = x The line through (k, ) and (7, 0) is perpendicular to the line = x PROBLEM SOLVING What is the distance between the lines = x and = x + 5? Verif our answer. 0. THOUGHT PROVOKING Find a formula for the distance from the point (x 0, 0 ) to the line ax + b = 0. Verif our formula using a point and a line.. COMPARING METHODS The point (x, ) lies on the line = x +. The distance from P(, 0) to (x, ) is represented b d. a. Write d as an expression in terms of x. b. Explain how ou can use the expression from part (a) to find the shortest distance from point P to the line = x +. c. Compare this method to the method used in Example. Which method do ou prefer? Explain our reasoning. Maintaining Mathematical Proficienc Plot the point in a coordinate plane. (Skills Review Handbook). A(, ). B(0, ). C(5, 0) 5. D(, ) Cop and complete the table. (Skills Review Handbook). x 0 = x Reviewing what ou learned in previous grades and lessons x 0 = x Chapter Parallel and Perpendicular Lines