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Math Help - distance between a point and a line.

  1. #1
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    distance between a point and a line.

    Show that the distance from a point (xZERO,yZERO) to a line Ax+By+C=0 is given by

    |AxZERO+ByZERO+C| / sqrt(A^2+B^2)

    Why absolute value? I get that hes putting the points into the slope of the other one to get maybe a parallel line. but thats all I get. why the sqrt of (a^2+b^2) ?
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  2. #2
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    Hello, frankinaround!

    \text}Show that the distance from a point }(x_o,y_o)\text{ to a line }Ax+By+C\:=\:0

    \text{is given by: }\;d \;=\;\dfrac{|Ax_o +By_o+C|}{\sqrt{A^2+B^2}}

    Why absolute value? . Because distance is a positive measure.

    I get that hes putting the points into the slope of the other one . What?
    to get maybe a parallel line, but thats all I get.

    Why the sqrt of (A^2+B^2) ? . Why not?

    You're expected to derive that formula, aren't you?

    Well, do it . . . and you'll see why.

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, frankinaround!


    You're expected to derive that formula, aren't you?

    Well, do it . . . and you'll see why.

    That's a bit harsh, the OP obviously does not know how to...

    Hint, the slope of the shortest line segment going through \displaystyle (x_0, y_0) and touching \displaystyle Ax + By + C = 0 will be perpendicular to the slope of \displaystyle Ax + By + C = 0.
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  4. #4
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    um... so im trying to do this. I can see ax/-b + c/-b = y and that the slop is a/-b, so then bx0/a should be the perpendicular slope. then I think I would need to take the point of intersection and use the distance formula to get to the next part of this question. but Im not sure how, because how do I know the point of intersection ? I have almost 2 equations. y=ax/-b+c/-b and bx/a + ? = y. I can TRY to substitute with y, so like bx/a + ? = ax/-b + c/-b. But it seems strange. what do you think?
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  5. #5
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    the equation of a line with slope \frac{B}{A} (perpendicular to the given line) through the point (x_0, y_0) is y= \frac{B}{A}(x- x_0)+ y_0. Where does that intersect y= -\frac{A}{B}x- \frac{C}{A}?
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