coordinate geometry formula: shortest distance between 2 lin

Expert replies

Previous Topic Next Topic

dinesh19aug: Senior | Next Rank: 100 Posts; Posts: 78; Joined: Thu Feb 21, 2008 9:05 am; Thanked: 7 times

by dinesh19aug » Sat Jul 31, 2010 1:31 pm

Hi,
Is there a direct formula to calculate/find the shortest distance between 2 lines??

Like this post

Source: Beat The GMAT — Problem Solving | Sat Jul 31, 2010 1:31 pm

sk818020: Master | Next Rank: 500 Posts; Posts: 214; Joined: Mon Mar 29, 2010 1:46 pm; Location: Houston, TX; Thanked: 37 times; GMAT Score:700

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

Like this post

dinesh19aug: Senior | Next Rank: 100 Posts; Posts: 78; Joined: Thu Feb 21, 2008 9:05 am; Thanked: 7 times

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

Like this post

sk818020: Master | Next Rank: 500 Posts; Posts: 214; Joined: Mon Mar 29, 2010 1:46 pm; Location: Houston, TX; Thanked: 37 times; GMAT Score:700

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

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

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.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

dinesh19aug: Senior | Next Rank: 100 Posts; Posts: 78; Joined: Thu Feb 21, 2008 9:05 am; Thanked: 7 times

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

Like this post

GMAT/MBA Expert

Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

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.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

Like this post

Audacio: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Sat Aug 06, 2011 4:17 pm

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?

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review