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coordinate geometry formula: shortest distance between 2 lin

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by sk818020 » Sat Jul 31, 2010 2:00 pm
To find the distance between point a and point b in a coordinate plane you use this formula.

Given two points (Xa, Ya), (Xb, Yb)

[(Xa-Xb)^2+(Ya-Yb)^2]

Hope that helps.

Thanks,

Jared
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by dinesh19aug » Sat Jul 31, 2010 2:11 pm
sk818020 wrote:To find the distance between point a and point b in a coordinate plane you use this formula.

Given two points (Xa, Ya), (Xb, Yb)

[(Xa-Xb)^2+(Ya-Yb)^2]

Hope that helps.

Thanks,

Jared
Jared,
This is not what I am looking for. I am looking for a formula to find the shortes distance between two LINES and NOT two POINTS.

Ex What is shortes distance beween two lines
1) 3y = 4x -3
2 y = 2x + 2
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by sk818020 » Sat Jul 31, 2010 2:15 pm
dinesh19aug wrote:
sk818020 wrote:To find the distance between point a and point b in a coordinate plane you use this formula.

Given two points (Xa, Ya), (Xb, Yb)

[(Xa-Xb)^2+(Ya-Yb)^2]

Hope that helps.

Thanks,

Jared
Jared,
This is not what I am looking for. I am looking for a formula to find the shortes distance between two LINES and NOT two POINTS.

Ex What is shortes distance beween two lines
1) 3y = 4x -3
2 y = 2x + 2
The way you asked the question makes the question unanswerable. The only time you could answer that question is if two lines are parallel to each other and not on top of each other. Then you could use the formula I gave by picking any two points where a perindicular line intersects with the original two lines to find out how far apart they are. If two lines are not parallel to each other, it would not make since to ask how far apart the two lines are because the distance between the two lines will vary the further away you get from the point at which they intersect (any two non-parallel lines will intersect just once).

Thanks,

Jared
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by GMATGuruNY » Sat Jul 31, 2010 4:24 pm
dinesh19aug wrote:
sk818020 wrote:To find the distance between point a and point b in a coordinate plane you use this formula.

Given two points (Xa, Ya), (Xb, Yb)

[(Xa-Xb)^2+(Ya-Yb)^2]

Hope that helps.

Thanks,

Jared
Jared,
This is not what I am looking for. I am looking for a formula to find the shortes distance between two LINES and NOT two POINTS.

Ex What is shortes distance beween two lines
1) 3y = 4x -3
2 y = 2x + 2
These two lines have different slopes; they are not parallel. Lines that are not parallel intersect, so the shortest distance between them would be 0, at the point of intersection.
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by dinesh19aug » Mon Aug 09, 2010 3:09 pm
GMATGuruNY wrote:
dinesh19aug wrote:
sk818020 wrote:To find the distance between point a and point b in a coordinate plane you use this formula.

Given two points (Xa, Ya), (Xb, Yb)

[(Xa-Xb)^2+(Ya-Yb)^2]

Hope that helps.

Thanks,

Jared
Jared,
This is not what I am looking for. I am looking for a formula to find the shortes distance between two LINES and NOT two POINTS.

Ex What is shortes distance beween two lines
1) 3y = 4x -3
2 y = 2x + 2
These two lines have different slopes; they are not parallel. Lines that are not parallel intersect, so the shortest distance between them would be 0, at the point of intersection.
Apologies _ i see that I confused many people with my example of two lines.
I just create a random equations , CONSIDER that the lines are parallel. I know that shortest distance between them is perpendicular line. I can solve for the distance, but it takes too long and it is cumbersome approach. I was looking for a direct formula. Any ideas
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by Ian Stewart » Mon Aug 09, 2010 4:27 pm
dinesh19aug wrote: Apologies _ i see that I confused many people with my example of two lines.
I just create a random equations , CONSIDER that the lines are parallel. I know that shortest distance between them is perpendicular line. I can solve for the distance, but it takes too long and it is cumbersome approach. I was looking for a direct formula. Any ideas
I didn't know a formula for this offhand, but you can work one out. If you have two parallel lines, they must have the same slope, so you'll be able to write their equations as follows:

y = mx + b
y = mx + c

Now y = (-1/m)x is perpendicular to both. We then just need to find where this perpendicular line intersects with each line above, and finally calculate the distance between the two intersection points. When I do that and simplify, I find that the distance d between the two points is:

d = |b-c|/sqrt(m^2 + 1)

I did that quickly, but I checked whether this works for a couple of lines, and it seems right.

In any case, while working out the formula isn't bad high-level coordinate geometry practice, you would never, ever need this formula on the GMAT. You may be asked to find the distance between two points on the test, but not between two lines.
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by Audacio » Sat Aug 06, 2011 4:23 pm
Ian Stewart wrote:
dinesh19aug wrote: Apologies _ i see that I confused many people with my example of two lines.
I just create a random equations , CONSIDER that the lines are parallel. I know that shortest distance between them is perpendicular line. I can solve for the distance, but it takes too long and it is cumbersome approach. I was looking for a direct formula. Any ideas
I didn't know a formula for this offhand, but you can work one out. If you have two parallel lines, they must have the same slope, so you'll be able to write their equations as follows:

y = mx + b
y = mx + c

Now y = (-1/m)x is perpendicular to both. We then just need to find where this perpendicular line intersects with each line above, and finally calculate the distance between the two intersection points. When I do that and simplify, I find that the distance d between the two points is:

d = |b-c|/sqrt(m^2 + 1)

I did that quickly, but I checked whether this works for a couple of lines, and it seems right.

In any case, while working out the formula isn't bad high-level coordinate geometry practice, you would never, ever need this formula on the GMAT. You may be asked to find the distance between two points on the test, but not between two lines.
Ian thank you very much for the formula for the shortest distance between 2 parrelel lines, the formula will help me in the future. Although I understand why you used |b-c| for the numerator, however I do not understand why you used sqrt(m^2+1) for the denominator would you please tell me why used that in the denominator?
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