DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Name Class 4.5 Date Equations of Parallel and Perpendicular Lines Essential Question: How can you find the equation of a line that is parallel or perpendicular to a given line? Explore Resource Locker Exploring Slopes of Lines Recall that the slope of a straight line in a coordinate plane is the of the rise to the run. In the figure, the slope of _ ratio rise 1 4 __ __ AB is ___ run = 8 = 2 . 3 y B(4, 1) 1 -5 -3 A(-4, -3) x -1 0 Run = 4-(-4) = 8 -5 A Graph the equations y = 2(x + 1) and y = 2x - 3. B What do you notice about the graphs of the two lines? About the slopes of the lines? The lines are parallel. The slopes are equal. © Houghton Mifflin Harcourt Publishing Company Use a protractor. What is the measure of the angle formed by the intersection of the lines. What does that tell you about the lines? 90°; the lines are perpendicular. D 2 x -4 -2 What are the slopes of the two lines? How are they related? 1 -__ and 3; the slopes are opposite reciprocals. Complete the statements: If two nonvertical lines are 0 2 4 -2 -4 y 8 6 3 E y 4 The graphs of x + 3y = 22 and y = 3x - 14 are shown. C Rise = 1-(-3) =4 3 4 2 x 0 2 4 6 8 parallel , then they have equal slopes. If two nonvertical lines are perpendicular, –1 then the product of their slopes is . Module 4 GE_MNLESE385795_U2M04L5.indd 205 205 Lesson 5 02/04/14 12:20 AM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A DO NOT Correcti Reflect 1. Your friend says that if two lines have opposite slopes, they are perpendicular. He uses the slopes 1 and –1 as examples. Do you agree with your friend? Explain. No; although lines with slopes of 1 and -1 are perpendicular, it’s because the product of the slopes is -1. Slopes of 2 and -2 are opposites, but the corresponding lines are not perpendicular. 2. The frets on a guitar are all perpendicular to one of the strings. Explain why 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. Writing Equations of Parallel Lines Explain 1 You can use slope relationships to write an equation of a line parallel to a given line. Example 1 Write the equation of each line in slope-intercept form. The line parallel to y = 5x + 1 that passes through (-1, 2) Parallel lines have equal slopes. So the slope of the required line is 5. y - y 1 = m(x - x 1) Use point-slope form. y - 2 = 5(x - (-1)) m, x 1,y 1. Substitute for y - 2 = 5x + 5 Simplify. Solve for y = 5x + 7 y. The equation of the line is y = -3x + 4 that passes through (9, -6) equal Parallel lines have Use point-slope form. m, x 1,y 1. Substitute for slopes. So the slope of the required line is -3 . y - y 1 = m(x - x 1) ( y - -6 = -3 x - 9 y + 6 = -3 x + 27 Simplify. Solve for The equation of the line is GE_MNLESE385795_U2M04L5.indd 206 ) y = -3 x + 21 y. Module4 © Houghton Mifflin Harcourt Publishing Company The line parallel to y = 5x + 7. y = -3x + 21 . 206 Lesson5 20/03/14 5:28 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Reflect 3. What is the equation of the line through a given point and parallel to the x-axis? Why? The equation is y = y 1 , wherey 1 is they-coordinate of the given point. This is because the x-axis is a horizontal line with equation y = 0. Your Turn Write the equation of each line in slope-intercept form. 4. The line parallel to y = -x that passes through (5, 2.5) 5. The line parallel to y = __32 x + 4 tha t passes through (-4, 0) _3 (x - (-4)) 2 3 y = _x + 6 y - (0) = y - 2.5 = -1(x - 5) y - 2.5 = -x + 5 2 y = -x + 7.5 Writing Equations of Perpendicular Lines Explain 2 You can use slope relationships to write an equation of a line perpendicular to a given line. Example 2 Write the equation of each line in slope-intercept form. The line perpendicular to y = 4x - 2 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 -__14 . © Houghton Mifflin Harcourt Publishing Company y - y 1 = m(x - x 1) Use point-slope form. 1 (x - 3)Substitutefor y - (-1) = -_ 4 3Simplify. 1x + _ y + 1 = -_ 4 4 1x - _ 1Solvefor y = -_ y. 4 4 The equation of the line is y = -__14 x - __14 . The line perpendicular to m, x 1,y 1. y = -__25 x + 12 that passes through (-6, -8) The product of the slopes of perpendicular lines is 5 . -1 . So the slope of the required line is __ 2 y - y 1 = m(x - x 1) Use point-slope form. 5 __ y - -8 = 2 y+8= y= 5 __ 2 5 __ 2 (x - -6 ) Substitute for m, x 1,y 1. x + 15 Simplify. x+ Solve for y. 7 5 x+ 7 . The equation of the line is y y = __ 2 Module4 GE_MNLESE385795_U2M04L5.indd 207 207 Lesson5 20/03/14 5:28 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A DO NOT Correcti 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. Why 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. 3 The line perpendicular to y = __ x + 2 that 2 passes through (3, –1) 8. The line perpendicular to y = -4x that passes through (0, 0) 1( x - 0) y - 0 = __ 4 2 (x - 3) y − (-1) = -__ 3 2 __ y = -3x + 1 1 y = __ x 4 Elaborate 9. 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? Explain. 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. Module 4 GE_MNLESE385795_U2M04L5.indd 208 208 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Zoran Zeremski/Shutterstock 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? Explain. No; if the point is on the line, the line can’t be parallel to the line, because it either Lesson 5 20/03/14 5:27 PM

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