CK-12 Foundation: 0505S Equations of Parallel and Perpendicular Lines (H264)
We can use the properties of parallel and perpendicular lines to write an equation of a line parallel or perpendicular to a given line. You might be given a line and a point, and asked to find the line that goes through the given point and is parallel or perpendicular to the given line. Here’s how to do this:
- Find the slope of the given line from its equation. (You might need to re-write the equation in a form such as the slope-intercept form.)
- Find the slope of the parallel or perpendicular line—which is either the same as the slope you found in step 1 (if it’s parallel), or the negative reciprocal of the slope you found in step 1 (if it’s perpendicular).
- Use the slope you found in step 2, along with the point you were given, to write an equation of the new line in slope-intercept form or point-slope form.
Investigate Families of Lines
Write the equation of the family of lines satisfying the given condition.
b) through the point (0, -1)
Watch this video for help with the Examples above.
CK-12 Foundation: Equations of Parallel and Perpendicular Lines
- A family of lines is a set of lines that have something in common with each other. Straight lines can belong to two types of families: one where the slope is the same and one where the y−intercept is the same.
- Notice that in such a family all the lines are parallel. All the lines look the same, except that they are shifted up and down the y−axis. As b gets larger the line rises on the y−axis, and as b gets smaller the line goes lower on the y−axis. This behavior is often called a vertical shift.
- Find the equation of the line parallel to 5x−2y=2 that passes through point (3, -2).
- Find the equation of the line perpendicular to y=−25x−3 that passes through point (2, 8).
- Find the equation of the line parallel to 7y+2x−10=0 that passes through the point (2, 2).
- Find the equation of the line perpendicular to y+5=3(x−2) that passes through the point (6, 2).
- Line S passes through the points (2, 3) and (4, 7). Line T passes through the point (2, 5). If Lines S and T are parallel, name one more point on line T. (Hint: you don’t need to find the slope of either line.)
- Lines P and Q both pass through (-1, 5). Line P also passes through (-3, -1). If P and Q are perpendicular, name one more point on line Q. (This time you will have to find the slopes of both lines.)
- Write the equation of the family of lines satisfying the given condition.
- All lines that pass through point (0, 4).
- All lines that are perpendicular to 4x+3y−1=0.
- All lines that are parallel to y−3=4x+2.
- All lines that pass through the point (0, -1).
- Name two lines that pass through the point (3, -1) and are perpendicular to each other.
- Name two lines that are each perpendicular to y=−4x−2. What is the relationship of those two lines to each other?
- Name two perpendicular lines that both pass through the point (3, -2). Then name a line parallel to one of them that passes through the point (-2, 5).