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Parallel and Perpendicular Lines

Learn to identify, graph, and calculate equations of parallel and perpendicular lines algebraically and visually, with practice exercises included.

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Parallel and Perpendicular Lines

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  1. Parallel and Perpendicular Lines

  2. Slope of Parallel Lines • Two non-vertical lines with the same slope are parallel.

  3. Parallel Lines • Two lines with the same slope are said to be parallel lines. If you graph them they will never intersect. • We can decide algebraically if two lines are parallel by finding the slope of each line and seeing if the slopes are equal to each other. • We can find the equation of a line parallel to a given line and going through a given point by: a.) first finding the slope m of the given line; b.) finding the equation of the line through the given point with slope m.

  4. Are the two lines:L1, through (-2, 1) and (4, 5)L2, through (3, 0) and (0, -2), parallel?

  5. Testing if Lines are Parallel Are the lines parallel? Find the slope of 12x + 3y = -9? 3y = -12x – 9 The slope m = -4 y = -4x -3 Find the slope of -8x -2y =14 -2y = 8x + 14 The slope m = -4 y = -4x – 7 Since the slopes are equal the lines are parallel.

  6. Graphs of Parallel Lines The red line is the graph of y = – 4x – 3 and the blue line is the graph of y = – 4x – 7

  7. Practice Testing if Lines are Parallel Are the lines parallel? (click mouse for answer) Since the slopes are different the lines are not parallel. parallel? (click mouse for answer) Are the lines Since the slopes are equal the lines are parallel.

  8. Parallel Remember parallel lines have the same slopes so if you need the slope of a line parallel to a given line, use the same slope.

  9. Constructing Parallel Lines Find the equation in standard form of a line going through the point (3, -5) and parallel to Using the point-slope equation where the slope m = 2/3 and the point is (3, -5) we get

  10. Practice Constructing Parallel Lines Find the equation of the line going through the point (4,1) and parallel to (click mouse for answer) Find the equation of the line going through the point (-2,7) and parallel to (click mouse for answer)

  11. Properties of Perpendicular Lines Perpendicular Lines What angles are formed by perpendicular lines? Right angles which equal 900.

  12. Slopes of Perpendicular Lines • If neither line is vertical, then the slopes of perpendicular lines are negative reciprocals. • If the product of the slopes of two lines is -1 then the lines are perpendicular. • Horizontal lines are perpendicular to vertical lines.

  13. Perpendicular Lines • Perpendicular lines are lines that intersect in a right angle. • We can decide algebraically if two lines are perpendicular by finding the slope of each line and seeing if the slopes are negative reciprocals of each other. This is equivalent to multiplying the two slopes together and seeing if their product is –1. • We can find the equation of a line perpendicular to a given line and going through a given point by: a.) first finding the slope m of the given line; b.) finding the equation of the line through the given point with slope = –1/m.

  14. Write parallel, perpendicular, or neither for the pair of lines that passes through (5, -9) and (3, 7) and the line through (0, 2) and (8, 3). You Try

  15. Testing if Lines Are Perpendicular Since the slopes are negative reciprocals of each other the lines are perpendicular.

  16. Graphs of Perpendicular Lines The red line is the graph of y = – 2x + 5 and the blue line is the graph of y = – 1/2 x +4

  17. Practice Testing if Lines Are Perpendicular Since the slopes are not negative reciprocals of each other (their product is not -1) the lines are not perpendicular Since the slopes are negative reciprocals of each other (their product is -1) the lines are perpendicular.

  18. Perpendicular Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal, and switch the sign to the opposite.

  19. Constructing Perpendicular Lines Find the equation in standard form of a line going through the point (3, -5) and perpendicular to Using the point-slope equation where the slope m = -3/2 and the point is (3, -5) we get

  20. Practice Constructing Perpendicular Lines Find the equation of the line going through the point (4,1) and perpendicular to (click mouse for answer) Find the equation of the line going through the point (-2,7) and perpendicular to (click mouse for answer)

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