Equations of Parallel and Perpendicular Lines

Equations of Parallel and Perpendicular Lines Study Notes

• A line parallel to (y = \frac{2}{3}x - 4) has a slope of (\frac{2}{3}); a perpendicular line has a slope of (-\frac{3}{2}).

• A line parallel to the y-axis is vertical and has an undefined slope, regardless of its passing point (e.g. through point ((-1, 5))).

• For the line defined by (4x - y = 6), parallel lines include:

  • (y + 1 = 4(x - 2))
  • (y = 4x + 11)
  • (8x - 2y = 6)

• To determine the slope-intercept form of a line parallel to (y = -\frac{4}{3}x + 11) and passing through point ((-6, 2)), the equation is given as (y = -\frac{4}{3}x - 6).

• The line (y = -8) is horizontal; a line perpendicular to it is vertical. The equation passing through point ((-4, -2)) is (x = -4).

• Lines are not perpendicular if their slopes are not opposite reciprocals.

• Triangle ABC is a right triangle if one side is perpendicular to another, confirming that segment (BC) is perpendicular to (AC).

• For quadrilateral with vertices (E(-4, 2)), (F(4, 7)), (G(8, 1)), and (H(0, -4)):

  • The slopes of (EF) and (GH) are both (\frac{5}{8}).
  • Quadrilateral (EFGH) is a parallelogram because both pairs of opposite sides are parallel.

• For the equations (-4 = 3x + 6y) and (ax - 8y = 12) to be parallel, (a) must equal (-4).

• Lines (y = 3x - 7) and (3x + 9y = 9) are perpendicular because their slopes (3 and (-\frac{1}{3})) are opposite reciprocals, yielding a product of (-1).