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High School Mathematics
9.13 Parallel Lines & Perpendicular Lines - Slope

Parallel lines:
  • Two lines are parallel if and only if they have the same slope. They never intersect.
  • If m1 is the slope of one line and m2 is the slope of another line then m1 = m2
Perpendicular lines:
  • Two lines are perpendicular if and only if the product of their slopes is - 1, or if one is vertical and the other horizontal.
  • They meet at a point to form right angles.
  • They have negative reciprocal slopes.
    The Product of the perpendicular lines slopes are equal to -1
    i.e., m1 and m2 are slopes of two perpendicular lines and their product m1m2 = -1
    i.e., m1 = -1/(m2)
Parallel lines Perpendicular lines
y = 2x + 5
y = 2x - 5
y = 2x + 7
y = 2x + 115
y = 4x + 11
y = -1/4x - 2
These lines are all parallel. They all have the same slope m = 2. These lines are perpendicular. Their slopes (m) are negative reciprocals.
Examples:
  1. Find the equation of the line passing through (5,4) and parallel to 3x - y - 15 = 0.
    solution:
    Any line parallel to 3x - y - 15 = 0 takes the form of 3x - y + k = 0.

    Note: For any equation of two parallel lines differ in constant terms, x and y coefficients being same.

    If 3x - y + k = 0 passing through (5,4), we have
    3.5 - 4 + k = 0
    15 - 4 + k = 0
    11 + k = 0
    Therefore, k = -11
    The required equation is 3x - y - 11 = 0.

  2. Find the equation of a line perpendicular to 3x - 4y + 1 = 0 and passing through (2,4).
    Solution:
    Given that,
    3x - 4y + 1 = 0, Slope of this equation is m1 = -(x-coefficient / y-coefficient).
    = -3/-4
    = 4/4
    Let the slope of the line perpendicular to 3x - 4y + 1 = 0 be m2
    We know that product perpendicular lines slopes are equal to -1
    i.e., m1m2 = -1
    We know m1 = 3/4
    (3/4)m2 = -1
    m2 = -4/3
    The slope of line required is -4/3 and a point on the line is (2,4).
    By the point slope form of a line we have,
    (y - y1) = m (x - x1)
    y - 4 = -4/3(x - 2)
    3y - 12 = -4x + 8
    4x + 3y - 20 = 0
    This is the required line.
Note:
The equation of x axis or horizontal line is y = 0. The slope is 0.
The equation of y axis or vertical line is x = 0. The slope is not defined.

Directions: Answer the following questions. Also write at least 10 examples of your own.
Q 1: The slope of a line perpendicular to the line whose equation is 3y + 2x = 6 is ____
Answer:

Q 2: The slope of one line is x/4 and the slope of another line is (x+5)/6. If the two lines are parallel then the value of x = ____.
Answer:

Q 3: The slope of one line is 3/4 and the slope of another line is 8/(x-6). If the two lines are perpendicular then the value of x = ____.
Answer:

Q 4: The slope of a line perpendicular to a line whose slope is -2/3 is ______
Answer:

Q 5: The slope of a line parallel to a line whose slope is -2/3 is ______
Answer:

Q 6: The slope of a line parallel to the line whose equation is 3y + 2x = 6 is ____
Answer:

Question 7: This question is available to subscribers only!

Question 8: This question is available to subscribers only!


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