In 2 earlier posts that I have done, I showed the students how to find an equation of a line (Finding equation of a line) and to determine how 2 lines will intersect (Determining if lines are parallel, perpendicular or not). In this post, I will show you how to find the equation of a line that is perpendicular to another line at a given point. In our example, we are looking for equation of line2 given some facts about line2 or the original line line1.
There are many variations of this type of problem, but at the end of the day, all you need to know is the slope of the original line, and any point on this perpendicular line. But here are some of the basic variations:
1. Given equation of line1, and a point on line2.
2. Given 2 points on line1, and a point on line2.
3. Given the slope of line1, and a point on line2.
I am going to use scenario #1 for our example. You are give that line1's equation is below, and a point on the perpendicular line line2 is (2,3).
line1: y = 2x + 1
Like I have stated earlier, all we need to do is to find the slope of the orginal line line1, and a point on this perpendicular line line2. If you refer back to the post on Determining if lines are parallel, perpendicular or not, you will recall that when 2 lines intersect perpendicularly (intersect with 90 degrees), the product of the slopes of the 2 lines will be equal to -1.
So given that line1 is y = 2x + 1. It tells us that the slope of this line is 2. We have to figure out the slope of the second line. Since (slope of line1) * (slope of line2) = -1, we can setup an equation to solve for slope of the second line.
(slope of line1) * (slope of line2) = -1
2 * (slope of line2) = -1
slope of line2 = -1/2
Now we know that the slope of the perpendicular line line2 is -1/2, and we are given a point on this line which is (2,3). Now you can refer to the first post that I have done on Finding equation of a line, and you can find the equation of this perpendicular line rather easily.
To complete this post, given slope of the perpendicular line is -1/2, and a point on this line is (2,3). We can use the slope intercept form of the equation to complete this task.
y = mx + b (substitute m = -1/2, and x = 2, and y = 3)
3 = -1/2(2) + b (solve for b, which is the y-intercept of the line)
3 = -1 + b
b = 4
Now you can rewrite the equation:
y = -1/2x + 4
And this is the equation of the line2 which is perpendicular to line which contains the point (2,3).
NOTE: The point on line2 DOES NOT have to lie on line1 (meaning that this given point does not need to be the intersecting point between line1 and line2. The reason is that as long as we know the slope of the line, and given any point on this line2, you can determine the equation of this line. The question asks for the equation of the line, but did not ask for the coordinate of the intersecting line between line1 and line2.
If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center. Or if you would like more examples on this type of problem, please feel free to reach out to us.
Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T: 201-444-8020
E: glenrock@mathnasium.com
www.mathnasium.com/glenrock
