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Writing equations of perpendicular lines

Given line A and point P, Sal finds the equation of the line perpendicular to A that passes through P. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Michael Lucas
    My question is, how do you find the intersecting point of the perpendicular lines if they do not intercept on the y axis? WITHOUT USING A GRAPH. I know this was touched on already but I think I am asking a slightly different question.

    For example, Find the distance between the point (2,6) and the line y=−2x.

    This tells me that the slope of the perpendicular line is 1/2. The equation for my second line would then be, (6) = 1/2*(2) + b. Solving this gives me the y-intercept of 5.

    My two points for my second line are now (2,6) (0,5). By using the graph(which is what I am trying to avoid), I can find that the intersecting point of both lines is (-2,4)

    Then, using the "distance formula" cough(pythagorean theorem)cough for the two points (2,6) and (-2,4) ==> (2-(-2))squared + (6-4)squared ==> 16+4 ==> square root of 20 being 2 root 5. So, I have the answer BY USING A GRAPH.

    My question is, how do you find the intersecting point of the perpendicular lines if they do not intercept on the y axis? WITHOUT USING A GRAPH

    What formula would be used? What if I end up not having graph paper? What if my line is screwy? I feel that having an equation to figure this out would let me know that I understand fully and NOT have to rely on NEEDING a graph to solve. Thanks for any help!
    (36 votes)
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    • aqualine ultimate style avatar for user Caroline
      Instead of finding the y-intercept of the point (2,6), find the complete equation of the line. Do this by doing what you did before--find your m and b values.

      You know that the line of (2,6) is perpendicular to y=-2x, so m=1/2. You already solved for b after substituting in the values of m, x and y:

      y=mx+b
      (6)=(1/2)(2)+b
      6-1=b, b=5

      But instead of finding your y intercept, write down the general equation of the line! Since you know m and b, the equation of this line is y=1/2x+5.

      Now what? To find where y=1/2x+5 and the original line y=-2x intersect, set them equal to each other. Let y in both of the equations equal the same value. You are doing this because at the two lines' point of intersection, both lines will share the same x and y value.

      So, let y=1/2x+5 equal y=-2x. That means

      -2x = 1/2x+5
      0= 5/2x +5
      -5 = 5/2x
      -2 =x

      Now you now that at the point where the two lines intersect, x=-2. You can substitute this x value into either of the two original equations to obtain the corresponding y value. Eg.

      Sub x=-2 into y=-2x

      y=-2(-2)
      y=4

      OR sub x=-2 into y=1/2x+5

      y=(1/2)(-2)+5
      =-1 + 5
      y=4

      Note that in both equations, you get the same y value.

      So now you know that the point of intersection is (-2,4), without graphing. I'm sure there's a video on this on the site; try looking up "substitution" or "elimination".
      (48 votes)
  • duskpin seed style avatar for user sanjana ponnam
    how would you solve this?

    Determine the equation of the line that is perpendicular to y= 3x+1 and has the same y-intercept.
    (10 votes)
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    • mr pink red style avatar for user andrewp18
      𝑦 = 3𝑥 + 1
      All perpendicular lines have negative reciprocal slopes of each other. Therefore, the slope of our perpendicular line would be -1/3. So our equation for our perpendicular line so far is:
      𝑦 = (-1/3)𝑥 + 𝑏
      𝑏 in this equation is simply the 𝑦-intercept. We are told that the perpendicular line has the same 𝑦-intercept as the original line so that means we have 𝑏 = 1. Therefore, our finally equation for our perpendicular line is:
      𝑦 = (-1/3)𝑥 + 1
      Comment if you have questions.
      (18 votes)
  • piceratops ultimate style avatar for user Pratik Patnaik
    Why the slope of a line B perpendicular to another line A is the negative inverse of Line A ?
    (10 votes)
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  • leaf red style avatar for user Nate
    Why is 1/2 the inverse of 2? How do you figure out the inverse of a number?
    (5 votes)
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  • piceratops ultimate style avatar for user alta mardell
    how do you find the equivalence in slopes
    (4 votes)
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    • piceratops ultimate style avatar for user bruzzf
      when you have the slope for let's say line A
      - then a line parallel to line A will have a slope equal to the slope of line A
      - a line perpendicular to line A will have a slope equal to the negative inverse of the slope of line A

      for example, if slope = 2
      then the negative inverse is -1/2
      (6 votes)
  • piceratops ultimate style avatar for user Gabriella
    How do I find the negative inverse of 1?
    (2 votes)
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    • hopper jumping style avatar for user Belinda Cheng
      Write your number as a fraction then flip the fraction and flip the sign. Hence, 1 could be written as 1/1. Flipping the fraction keeps it the same as 1/1. Change the positive sign to a negative sign, so it's -1/1 or -1. Here's another example. If you have a slope of -1/4, flip it to -4/1 then change the sign, making the perpendicular slope 4/1 or 4.
      (4 votes)
  • female robot grace style avatar for user Lali
    I've been working out the follow up problems to this video and I'm stuck at finding the distance between (0,-8) and x = 5. What do I do when I get an equation that is not clearly y = mx + b?
    (1 vote)
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    • old spice man green style avatar for user jmascaro
      Hi Alicia,
      x=5 means that for all values of y, x = 5. So on the graph, there will be a vertical line at x=5. To figure out the distance, then, from (0,-8) to x=5, draw a perpendicular line towards x=5. In this case, that line will be horizontal and the lines will meet at (5,-8). Since your x value at (0,-8) is 0, and your x value at the intersection (5,-8) is 5, and your y value remains the same, the distance between (0,-8) and x=5 is 5.
      Hope that helps.
      (6 votes)
  • blobby green style avatar for user kydriel.smith
    my question is how to add them up.
    (3 votes)
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  • blobby green style avatar for user Rania Tariq
    Is the sign also going to be inversed?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      Two things, perpendicular slopes have opposite (signs) reciprocal (multiplicative inverse). So if you have a slope of -3/2, the sign is changed to positive 2/3, no sign to invert. If you have 3/2, the answer is -2/3, the negative sign is added, but there is no sign to invert either. Secondly, negatives drift to the top, so even if you were not talking about perpendicular lines, if you want the multiplicative inverse of -2/3, you could say 3/-2,but then the negative would drift to the top to get -3/2.
      (2 votes)
  • duskpin seedling style avatar for user nick
    what about fractions? my equation; line a is y= -4/5x-19 with line b having points (4,-3) and the two are perpendicular
    (2 votes)
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    • mr pink green style avatar for user David Severin
      Perpendicular lines have opposite (signs) reciprocal (flip fraction). So if slope is -4/5, the perpendicular slope is 5/4. If y=mx+b, we can subtract mx on both sides to get b=y-mx, and subbing in x=4, y=-3, and m=5/4, we get b=-3-(5/4)(4), 4s cancel, so you have b=-3-5=-8. Thus equation is y=5/4 x -8.
      (1 vote)

Video transcript

We're asked what is the equation of line B? And they tell us that line A has an equation y is equal to 2x plus 11. And they say that the line B contains the point 6, negative 7. And they tell us lines A and B are perpendicular, so that means that slope of B must be negative inverse of slope of A. So what we'll do is figure out the slope of A, then take the negative inverse of it. Then we'll know the slope of B, then we can use this point right here to fill in the gaps and figure out B's y-intercept. So what's the slope of A? This is already in slope-intercept form. The slope of A is right there, it's the 2, mx plus b. So the slope here is equal to 2. So the slope of A is 2. What is the slope of B? So what is B's slope going to have to be? Well, it's perpendicular to A, so it's going to be the negative inverse of this. The inverse of two is 1/2. The negative inverse of that is negative 1/2. So B's slope is negative 1/2. So we know that B's equation has to be y is equal to its slope, m times x plus some y-intercept. We still don't know what the y-intercept of B is, but we can use this information to figure it out. We know that y is equal to negative 7 when x is equal to 6. Negative 1/2 times 6 plus b, right? I just know that this is on the point, so this point must satisfy the equation of line B. So let's work out what b must be-- or what b, the y-intercept, this is a lowercase b, not the line B. So we have negative 7 is equal to-- what's negative 1/2 half times 6? That's not a b there, that's a 6. What's negative 1/2 times 6? It's negative 3, is equal to negative 3 plus our y-intercept. Let's add 3 to both sides of this equation, so if we add 3 to both sides-- I just want to get rid of this 3 right here-- what do we get? The left-hand side, negative 7 plus 3 is negative 4, and that's going to be equal to-- these guys cancel out-- that's equal to b, our y-intercept. So this right here is a negative 4. So the equation of line B is y is equal to-- its slope is a negative inverse of this character-- so negative 1/2, negative 1/2 x. And its y-intercept we just figured out is negative 4. And we are done.