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Parallel and Perpendicular Lines

Chapter 6 Coordinate Geometry. 6.7. Parallel and Perpendicular Lines. 6.7. 1. MATHPOWER TM 10, WESTERN EDITION. Parallel Lines. B(0, 5). D(3, 0). A(-3, 0). If the slopes of two lines are equal , the lines are parallel. C(0, -5). If two lines are parallel ,

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Parallel and Perpendicular Lines

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  1. Chapter 6 Coordinate Geometry 6.7 Parallel and Perpendicular Lines 6.7.1 MATHPOWERTM 10, WESTERN EDITION

  2. Parallel Lines B(0, 5) D(3, 0) A(-3, 0) If the slopes of two lines are equal, the lines are parallel. C(0, -5) If two lines are parallel, their slopes are equal. AB is parallel to CD. 6.7.2

  3. Verifying Parallel Lines Show that the line segment AB with endpoints A(2, 3) and B(6, 5) is parallel to the line segment CD with endpoints C(-1, 4) and D(3, 6). Since the slopes are equal, the line segments are parallel. 6.7.3

  4. Using Parallel Slopes to Find k The following are slopes of parallel lines. Find the value ofk. 2k = 12 k = 6 -1k = 10 k = -10 -7k = -6 -2k = 15 k = k = 6.7.4

  5. Perpendicular Lines D(-1, 4) B(4, 2) If the slopes of two lines are negative reciprocals, the lines are perpendicular. C(3, -2) A(-2, -2) AB is perpendicular to CD. If two lines are perpendicular, their slopes are negative reciprocals. 6.7.5

  6. Perpendicular Line Segments Show that the line segment AB with endpoints A(0, 2) and B(-3, -4) is perpendicular to the line segment CD with endpoints C(2, -4) and D(-8, 1). The slopes are equal so line segments are perpendicular. 6.7.6

  7. Using Perpendicular Slopes to Find k The following are slopes of perpendicular lines. Find the value ofk. -5k = -2 -3k = 8 k = k = -3k = -10 -2k = 21 k = k = 6.7.7

  8. Parallel and Perpendicular Lines Given the following equations of lines, determine which are parallel and which are perpendicular. A) 3x + 4y - 24 = 0 B) 3x - 4y + 10 = 0 C) 4x + 3y - 16 = 0 D) 6x + 8y + 15 = 0 -4y = -3x - 10 y = x + 5/2 4y = -3x + 24 y = x + 6 Slope = Slope = 8y = -6x - 15 3y = -4x + 16 Slope = Slope = Lines A and D have the same slope, so they are parallel. Lines B and C have negative reciprocal slopes, so they are perpendicular. 6.7.8

  9. Writing the Equation of a Line Find the equation of the line through the point A(-1, 5) and parallel to 3x - 4y + 16 = 0. Find the slope. y - y1 = m(x - x1) 3x - 4y + 16 = 0 -4y = - 3x - 16 y - 5 = (x --1) y = x + 4 4y - 20 = 3(x + 1) 4y - 20 = 3x + 3 0 = 3x - 4y + 23 Slope = 3x - 4y + 23 = 0 6.7.9

  10. Writing the Equation of a Line Find the equation of the line through the point A(-1, 5) and perpendicular to 3x - 4y + 16 = 0. Find the slope. 3x - 4y + 16 = 0 -4y = -3x - 16 y - y1 = m(x - x1) y - 5 = (x --1) y = x + 4 3y - 15 = -4(x + 1) 3y - 15 = -4x - 4 4x + 3y - 11 = 0 Slope = Therefore, use the slope 4x + 3y - 11 = 0 6.7.10

  11. Writing the Equation of a Line Determine the equation of the line parallel to 3x + 6y - 9 = 0 and with the samey-intercept as 4x + 4y - 16 = 0. 3x + 6y - 9 = 0 6y = -3x + 9 4x + 4y - 16 = 0 For the y-intercept, x = 0: 4(0) + 4y - 16 = 0 4y = 16 y = 4 . The slope is . A point is (0, 4). y - y1 = m(x - x1) y - 4 = (x - 0) 2y - 8 = -1x x + 2y - 8 = 0 6.7.11

  12. Writing the Equation of a Line Determine the equation of the line that is perpendicular to 3x + 6y - 9 = 0 and has the same x-intercept as 4x + 4y - 16 = 0. 3x + 6y - 9 = 0 6y = -3x + 9 4x + 4y - 16 = 0 For the x-intercept, y = 0: 4x + 4(0)- 16 = 0 4x = 16 x = 4 The slope is 2. A point is (4, 0). y - y1 = m(x - x1) y - 0 = 2(x - 4) y = 2x - 8 0 = 2x - y - 8 The equation of the line is 2x - y - 8 = 0. 6.7.12

  13. Writing the Equation of a Line Determine the equation of each of the following lines. A) perpendicular to 5x - y - 1 = 0 and passing through (4, -2) x + 5y + 6 = 0 B) perpendicular to 2x - y - 3 = 0 and intersects the y-axis at -2 x + 2y + 4 = 0 C) parallel to 2x + 5y + 10 = 0 and same x-intercept as 4x + 8 = 0 2x + 5y + 4 = 0 D) passing through the point (3, 6) and parallel to the x-axis y = 6 or y - 6 = 0 E) passing through the y-intercept of 6x + 5y + 25 = 0 and parallel to 4x - 3y + 9 = 0 4x - 3y - 15 = 0 F) passing through the x-intercept of 6x + 5y + 30 = 0 and perpendicular to 4x - 3y + 9 = 0 3x + 4y + 15 = 0 6.7.13

  14. Assignment Suggested Questions: Pages 294 and 295 1 - 25 odd, 27ace, 28 - 42 even, 44 - 50 6.7.14

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