perpendicular distance from point B(0,5) to the line x+3y-5=0

- anonymous

perpendicular distance from point B(0,5) to the line x+3y-5=0

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- ash2326

@virtus Are you here?

- anonymous

yes!

- ash2326

@virtus There maybe a direct formula for this, but I don't remember that. I'll start from the basics. Do you have time for this?

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- anonymous

sure, thanks ash2326

- anonymous

Can I give the formula??

- ash2326

Thanks virtus, let's begin. I recommend that you understand the problem, you'll remember it always. What do you want?

- anonymous

Go ahead @ash2326 ..

- ash2326

|dw:1341645333105:dw|
Let AB be our line and C is the point for which we need to find the perpendicular distance.
Do you understand the diagram?

- anonymous

yes

- ash2326

You can see that the perpendicular line from the point C is intersecting the line AB at a point. Let it be D
If we could find the coordinates of D, then we can easily find the distance.
do you have any ideas what should we do?

- anonymous

POI

- ash2326

POI???

- anonymous

find the Point Of Intersection

- anonymous

Ha ha ha ha..

- ash2326

Yeah:) @virtus
For that we need to find the equation of the line CD, do you have any idea how to do that?

- anonymous

well you know what point C is and the gradient of line CD would be -1/ gradient of line AB

- anonymous

oh wait do we know what C is ?

- ash2326

Awesome:D Could you do that?
We have AB
as
\(x+3y-5=0\)
Yeah we know C as (0, 5)

- anonymous

x+3y-15=0

- ash2326

Oops, You made a mistake
I can see that perpendicular line's slope is same as AB, it's to be -1/ slope AB
Check again :)

- anonymous

oh silly me

- ash2326

No problem, :D
find it again?

- anonymous

y-5 =3(x-0)
y=3x +5
3x-y+5=0

- ash2326

Correct:D Now find the intersection of the line (3x-y+5=0) and (x+3y-5=0)
this will give us D

- anonymous

D (-1,2)

- ash2326

Great work:D
now find the distance between (0, 5) and (-1, 2) using distance formula.

- ash2326

That will be the perpendicular distance:D

- anonymous

ngawwwwwww! I SEE!!!! so that's how you do it ;)
oh btw i got square root 10

- ash2326

Yeah, that's correct:D Great work btw:D

- anonymous

thanks so much for your invaluble help @ash2326 :D

- ash2326

@virtus you did all the work. I just guided you:D You are a good student:D

- anonymous

you can also use the formula from analytic geometry that the distance from a given point to a given line in general form \(\large Ax+By+C=0 \) is given by
Distance = \(\large \frac{Ax_0+By_0+C}{\pm \sqrt{A^2+B^2}} \) where \(\ (x_0, y_0) \) are the coordinates of the given point.

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