How to use Algebra to find parallel and perpendicular lines. Show Parallel LinesHow do we know when two lines are parallel? Their slopes are the same! Example:Find the equation of the line that is:
The slope of y=2x+1 is: 2 The parallel line needs to have the same slope of 2. We can solve it using the "point-slope" equation of a line: y − y1 = 2(x − x1) And then put in the point (5,4): y − 4 = 2(x − 5) And that answer is OK, but let's also put it in y = mx + b form: y − 4 = 2x − 10 y = 2x − 6 Vertical LinesBut this does not work for vertical lines ... I explain why at the end. Not The Same LineBe careful! They may be the same line (but with a different equation), and so are not parallel. How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope: Example: is y = 3x + 2 parallel to y − 2 = 3x ?For y = 3x + 2: the slope is 3, and y-intercept is 2 For y − 2 = 3x: the slope is 3, and y-intercept is 2 In fact they are the same line and so are not parallel Perpendicular LinesTwo lines are Perpendicular when they meet at a right angle (90°). To find a perpendicular slope: When one line has a slope of m, a perpendicular line has a slope of −1m In other words the negative reciprocal Example:Find the equation of the line that is
The slope of y=−4x+10 is: −4 The negative reciprocal of that slope is: m = −1−4 = 14 So the perpendicular line will have a slope of 1/4: y − y1 = (1/4)(x − x1) And now put in the point (7,2): y − 2 = (1/4)(x − 7) And that answer is OK, but let's also put it in "y=mx+b" form: y − 2 = x/4 − 7/4 y = x/4 + 1/4 Quick Check of PerpendicularWhen we multiply a slope m by its perpendicular slope −1m we get simply −1. So to quickly check if two lines are perpendicular: When we multiply their slopes, we get −1 Like this: Are these two lines perpendicular?
When we multiply the two slopes we get: 2 × (−0.5) = −1 Yes, we got −1, so they are perpendicular. Vertical LinesThe previous methods work nicely except for a vertical line: In this case the gradient is undefined (as we cannot divide by 0): m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined So just rely on the fact that:
Summary
What is the slope of a line parallel to this line?The slope of the parallel line is 0 and the slope of the perpendicular line is undefined.
|