Collinear Points -- Ways to determine if points are collinear

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  • #1
nycmathguy
Homework Statement:
Collinear Points Discussion
Relevant Equations:
Lines & Collinear Points
Chapter 1, Section 1.1
Collinear Points

59. Three or more points are collinear
when they all lie on the same line. Use the steps below to determine whether the set of points {A(2, 3), B(2, 6),C(6, 3)} and the set of points {A(8, 3), B(5, 2), C(2, 1)} are collinear.

(a) For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship exists among these distances for each set of points?

(b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line?

(c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity.

Let me see.

For (a), the Distance Formula must be used several times for each set.

Yes?

For (b), we simply plot each point on the xy-plane. We then see if the points are on the same line.

Yes?

For (c), you say? How do we use the Distance Formula to determine collinearity?
 

Answers and Replies

  • #2
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Homework Statement:: Collinear Points Discussion
Relevant Equations:: Lines & Collinear Points

Chapter 1, Section 1.1
Collinear Points

59. Three or more points are collinear
when they all lie on the same line. Use the steps below to determine whether the set of points {A(2, 3), B(2, 6),C(6, 3)} and the set of points {A(8, 3), B(5, 2), C(2, 1)} are collinear.

(a) For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship exists among these distances for each set of points?

(b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line?

(c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity.

Let me see.

For (a), the Distance Formula must be used several times for each set.

Yes?
Well, of course. That's what the instructions tell you to do.
nycmathguy said:
For (b), we simply plot each point on the xy-plane. We then see if the points are on the same line.

Yes?
Again, that's just a paraphrase of what the instructions say to do.
nycmathguy said:
For (c), you say? How do we use the Distance Formula to determine collinearity?
Let AC be the distance from A to C, AB the distance from A to B, and BC the distance from B to C. If the sum of the shorter distances is equal to the longer distance, the three points are collinear.

Another way to do it, but different from where they're steering you, is to calculate the slope from A to C and the slope from C to B. If they're equal, the three points are collinear. This might not be kosher if the book hasn't discussed slope yet.
 
  • #3
nycmathguy
Well, of course. That's what the instructions tell you to do.
Again, that's just a paraphrase of what the instructions say to do.
Let AC be the distance from A to C, AB the distance from A to B, and BC the distance from B to C. If the sum of the shorter distances is equal to the longer distance, the three points are collinear.

Another way to do it, but different from where they're steering you, is to calculate the slope from A to C and the slope from C to B. If they're equal, the three points are collinear. This might not be kosher if the book hasn't discussed slope yet.

This is a Ron Larson book Edition 10E.
 
  • #4
symbolipoint
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Homework Statement:: Collinear Points Discussion
Relevant Equations:: Lines & Collinear Points

Chapter 1, Section 1.1
Collinear Points

59. Three or more points are collinear
when they all lie on the same line. Use the steps below to determine whether the set of points {A(2, 3), B(2, 6),C(6, 3)} and the set of points {A(8, 3), B(5, 2), C(2, 1)} are collinear.

(a) For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship exists among these distances for each set of points?

(b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line?

(c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity.

Let me see.

For (a), the Distance Formula must be used several times for each set.

Yes?

For (b), we simply plot each point on the xy-plane. We then see if the points are on the same line.

Yes?

For (c), you say? How do we use the Distance Formula to determine collinearity?
This is an exercise which needs to be well understood through study of "Introductory Algebra", or Algebra 1. Once someone has sufficiently studied and successfully pass Algebra 1, the currently presented exercise is no longer a topic of difficulty.

Upon looking at other details of your exercise, it seems some knowledge of Intermediate Algebra might be needed. Let me alter the response paragraph:

This is an exercise which needs to be well understood through study of "Intermediate Algebra", or Algebra 2. Once someone has sufficiently studied and successfully pass Algebra 2, the currently presented exercise is no longer a topic of difficulty is a typical chore for study of the relevant concepts and skills, leading to their stronger understanding.
 
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  • #5
symbolipoint
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This is a Ron Larson book Edition 10E.
I might have misunderstood some of the nature of your currently asked Coordinate Geometry question.

What is the title of the Larson textbook?
 
  • #6
nycmathguy
I might have misunderstood some of the nature of your currently asked Coordinate Geometry question.

What is the title of the Larson textbook?
 

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  • #7
symbolipoint
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Homework Statement:: Collinear Points Discussion
Relevant Equations:: Lines & Collinear Points

For (c), you say? How do we use the Distance Formula to determine collinearity?
Also, thanks for showing us the title of your book.

Hopefully you obviously understand that the Distance Formula does not give information about collinearity of points. It is, of course only a way to find DISTANCE between two points. If you wish to know if three or more points IN THE PLANE are on the same line, Use formula for Slope. If ALL the point combinations in your set of points have the same slope, then the points are collinear.
 
  • #8
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Hopefully you obviously understand that the Distance Formula does not give information about collinearity of points.
The distance formula can be used to check collinearity. Given point A, B, and C, with B between A and C, if d(AB) + d(BC) = d(A, C), the points are on the same line.
 
  • #9
nycmathguy
I want to thank Delta2 for liking all my comments and threads. It is encouraging to know that at least one person here does not have something negative to say about a simple, middle-aged man living a humdrum life in NYC trying to advance, mathematically speaking.

When I found out the other day that I have an enlarged prostate, fear tried to cripple my hope to become a somewhat decent, amateur math guy. The first thought that comes to mind is cancer. It may or may not be the case, but the thought did cross my mind. So, diving into precalculus and calculus 1 helps me to forget my troubles (at least while solving for x). Again, thank you Delta2.
 
  • #10
FactChecker
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We can only give hints to help you solve homework problems. I think that the main problem is that we expect more effort to answer the problem than just restating it. Don't be discouraged or upset. Put down some equations and calculations, then we can help.
 
  • #11
nycmathguy
We can only give hints to help you solve homework problems. I think that the main problem is that we expect more effort to answer the problem than just restating it. Don't be discouraged or upset. Put down some equations and calculations, then we can help.

Allow me to correct you by asking one question:

WHAT HOMEWORK PROBLEMS?

I am not a student, sir. I am 56 years old. I have several math textbooks in my room. I love answering math questions just like some people enjoy puzzles. There is no test to prepare for. No teacher waiting for me to arrive the morning.
 
  • #12
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Allow me to correct you by asking one question:

WHAT HOMEWORK PROBLEMS?

I am not a student, sir. I am 56 years old. I have several math textbooks in my room. I love answering math questions just like some people enjoy puzzles. There is no test to prepare for. No teacher waiting for me to arrive the morning.
None of this matters here at PF. If the problem is in a textbook, we consider it to be a homework problem, independent of whether you are a student or are taking a class.
 
  • #13
nycmathguy
None of this matters here at PF. If the problem is in a textbook, we consider it to be a homework problem, independent of whether you are a student or are taking a class.
I welcome all hints.
I welcome any help you can give.
I welcome anything that is positive.
 
  • #14
FactChecker
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I welcome all hints.
I welcome any help you can give.
I welcome anything that is positive.
Then you must show more work than just restating the problem.
I see that you have since posted other questions that are much better.
 
  • #15
symbolipoint
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None of this matters here at PF. If the problem is in a textbook, we consider it to be a homework problem, independent of whether you are a student or are taking a class.
This is obviously a matter of interpretation and both sides need to be understood. When one studies on his own outside of formal course attendance, all of his work is homework.
 
  • #16
nycmathguy
Then you must show more work than just restating the problem.
I see that you have since posted other questions that are much better.
Show more work? That's all I do here is show math work using my cell phone.
 
  • #17
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Show more work? That's all I do here is show math work using my cell phone.
FactChecker's comment was about some of your other posts where you set up a problem, but don't follow through with a calculation.
 
  • #18
nycmathguy
FactChecker's comment was about some of your other posts where you set up a problem, but don't follow through with a calculation.
Mark,

I sometimes ask for help setting up a problem or ask if my set up is right. I then work out the problem on paper, especially if the problem requires a long calculation and time is short. It's 6:15 pm in NYC. I gotta go back to my job to work 9pm to 7am. Yes, I have two breaks at the job but solving math problems at the job on my break is very different than solving math problems on my days off.
 
  • #19
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I sometimes ask for help setting up a problem or ask if my set up is right.
And what some have said (and some have expressed frustration at) is why don't you finish the problem, and then check it, for those types of problems where a check makes sense. In several of your problems, just setting it up is at least 75% of the work needed. If your solution doesn't satisfy the problem statement, then that's where we can help out.
 
  • #20
symbolipoint
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Show more work? That's all I do here is show math work using my cell phone.
That is definitely an obstacle.

You should or want to write your preparations on paper with pen or pencil. You might in some cases use some computerized wordprocessing application or feature and make use either of any typesetting capability for Mathematical expressing; maybe whatever drawing program that you can use for making diagrams or figures; and if something is online which you want to see or view or read, DO THIS USING A 'NORMAL' COMPUTER ---- NOT a cellular phone.
 
  • #21
symbolipoint
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if my set up is right. I then work out the problem on paper, especially if the problem requires a long calculation and time is short.
GOOD! I did not yet read further posts while I gave my earlier response.
 

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