Geometry

Lines in the Coordinate Plane

Big Picture

In addition to having many properties involving transversals, parallel and perpendicular lines also have specialrelationships on the coordinate plane involving slope. Two parallel lines always have the same slope, and perpendicularlines have slopes that are negative reciprocals of each other. Equations can be written in slope-intercept form to makeit easier for us to graph them and find their slopes.

Key Terms

Slope: The steepness of a line, usually denoted by m.

y-Intercept: The point where the line crosses the y-axis.

Slopes of Lines

Given two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line: slope = \(m = \frac{rise}{run} = \frac{y_2 - y_1}{x_2 - x_1}\)

  • Positive slope: rises from left to right
  • Negative slope: falls from left to right
  • Zero slope: horizontal
  • Undefined: vertical
  • Since the denominator for the slope is equal to 0, the slope for a vertical line is undefined.

Parallel and Perpendicular Lines

Parallel Lines

Postulate: If two lines are parallel, they have the sameslope and different y-intercepts.

  • Any two vertical lines are parallel.

Perpendicular Lines

Postulate: If two lines are perpendicular, their slopes are reciprocals of each other (e.g. 1⁄2 and -2). The product of their slopes is -1.

  • Horizontal lines are perpendicular to vertical lines.

Geometry

Lines in the Coordinate Plane Cont.

Equations of Lines

Slope-intercept form: \(y = mx+b\), where \(m\) is the slope and \(b\) is the y-intercept

Standard form: \(Ax + By = C\), where \(A, B\), and \(C\) are constants and \(A\) and \(B\) are nonzero

If given a graph of a line, the equation of the line can be found using the slope-intercept form:

  • Find the slope - this is \(m\).
  • Find the \(y-\) intercept - this is \(b\).
  • Write the equation.

Graphing Parallel and Perpendicular Lines

If asked to draw a line parallel or perpendicular to a given line that goes through a given point:

  • Find the slope of the given line.
  • Find the slope (\(m\)) of the new line and plug it into the slope-intercept form of the equation.
  • Plug in the coordinates of the given point and solve to find the equation.

The steps to draw the perpendicular bisector of a given line segment are similar.

  • Find the midpoint of the line segment. (Recall that the perpendicular bisector goes through the midpoint of the line segment.)
  • Find the slope of the segment.
  • Find the perpendicular slope and plug it into the slope-intercept form of the equation.
  • Plug in the coordinates of the given point and solve to find the equation.

Shortest Distance

The shortest distance between a point and a line: the length of the segment starting at the point that’s perpendicular to the line.

  • Find the slope of the given line.
  • Find the perpendicular slope and plug it into the slope-intercept form of the equation.
  • Plug in the coordinates of the given point and solve to find the equation.
    To calculate the distance, there are a few more steps to follow:
  • Find where the perpendicular line intersects the given line. This can either be done by graphing or by setting the equations of the line and perpendicular line equal to each other and solving for the point of intersection.
  • Plug the coordinates of the given point and the point of intersection into the distance formula.

The shortest distance between two parallel lines is found in a similar way. The shortest distance is the length of the perpendicular segment that cuts between the two lines.

  • It doesn’t matter which perpendicular line is chosen - there are infinitely many perpendicular lines between two parallel lines.

Notes