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Parallel and Perpendicular lines

Parallel and Perpendicular lines. Parallel Lines = two different lines with the same slope---they run next to each other. All vertical lines are parallel. All horizontal lines are parallel.

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Parallel and Perpendicular lines

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  1. Parallel and Perpendicular lines

  2. Parallel Lines = two different lines with the same slope---they run next to each other. All vertical lines are parallel. All horizontal lines are parallel. • Perpendicular Lines = two lines that have slopes m and -1/m----they are lines that form right angles with each other. Slopes are negative reciprocals. All vertical lines are perpendicular to all horizontal lines. All horizontal lines are perpendicular to all vertical lines.

  3. Parallel and Perpendicular The slope is a number that tells "how steep" the line is and in which direction. So as you can see, parallel lines have the same slopes so if you need the slope of a line parallel to a given line, simply find the slope of the given line and the slope you want for a parallel line will be the same. 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 (flip it over) and make it negative.

  4. 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.

  5. Testing if Lines are Parallel Are the lines parallel? Find the slope of The slope m = -4 The slope m = -4 Find the slope of Since the slopes are equal the lines are parallel.

  6. 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.

  7. Write the equation of the line parallel the line 4x – 5y = 7 that passes through the point (-3, 7) What is the slope of the given line? y – y1 = m(x – x1)

  8. Constructing Parallel Lines Find the equation 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

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

  10. Write an equation in slope-intercept form of a line parallel to y = 3x – 7 with a y-intercept of 4 • Write an equation in slope-intercept form of a line parallel to y = .5x + 5 with a y-intercept of -2 • Write an equation in slope-intercept form for the line that contains the point (-3, -4) and is parallel to the graph of y = -4x - 2

  11. 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.

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

  13. 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.

  14. Write the equation of the line perpendicular the line 3x + 2y = 9 that passes through the point (2, 5) What is the slope of the given line? y – y1 = m(x – x1)

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

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

  17. Write an equation in slope-intercept form of a line perpendicular to y = 3x + 2 with a y-intercept of 4 • Write an equation in slope-intercept form of a line perpendicular to y = 4x + 2 with a y-intercept of 6 • Write an equation in slope-intercept form of a line perpendicular to y = -2x + 3 with a y-intercept of -5

  18. Write an equation in point-slope form for a line that contains the point (4, 5) and that is perpendicular to the line 2x + 3y = 7 • Write an equation in point-slope form for a line that contains the point (4, -3) and that is perpendicular to the line -10x + 2y = 8 • Write an equation in point-slope form for a line that contains the point (3, -2) and that is perpendicular to the line 4x – 2y = -6

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