How many lines can 5 dots create in geometry?

[Physicsissuef]

Homework Statement

5 dots in some plane can create how many lines?

a)5

b)10

c)15

d)20

Homework Equations

The Attempt at a Solution

If you ask me, I will say infinite, but the creators of the question think that one line can be created from at least 2 dots. Line can be created from 3, 4 ... n dots.

2 dots can create C_5^2 lines, and it is 10 lines. But what about those with 3 dots?

 

[malawi_glenn]

why infinite number of lines?

I think you should just take 5 random points, and then count the number of lines.

 

[HallsofIvy]

Perhaps it would be better to say 5 points. A 'dot' is not a mathematical object. Of course, if the 5 points happened to lie on a single line, there would be only one line through them so the question really should be "What is the largest number of lines that can be drawn through 5 points in a plane?"

Choosing any point, you can draw a line through it and anyone of the 4 other points. That is, each point lies on 4 lines. If we multiply the number of points by the number of lines "per" point, we have 4(5)= 20. But each line accounts for 2 points so the correct answer is 20/2= 10.

In fact, the largest number of lines that can be draw through n points in a plane is n(n-1)/2.

Notice by the way that we can argue this in a slightly different way: choose one point (call it p1) out of the n points. You can draw lines through it and each of the other n-1 points: n-1 lines. Now pick another point (p2)( and draw lines through it and each of the other points, NOT counting p1. There are n- 2 "other points" (not p1 or p2) so n-2 such lines. Pick a third point, p3, and we can draw n- 3 lines through n-3 "other points" (not p1, p2, or p3). We can go through all of the points until we have used them all up and have (n-1)+ (n-2)+ (n-3)+ ...+ 3+ 2 +1 lines. That is the sum of the first n-1 positive integers which is well know to be n(n-1)/2. (It is perhaps better known that the sum of the first n integers is n(n+1)/2 but that immediately gives the first.)

 

[Physicsissuef]

HallsofIvy yes, I understand. But what I do not understand is that also line can be created by three points, or in that case the line will be repeated?

 

[HallsofIvy]

Yes, if three points happen to lie on a line the method I cite would count that same line three times. That was why I said the question should be "What is the largest number of lines that can be drawn through 5 points".

 

[Physicsissuef]

Yes. Ok, thank you. And what about this one:

Five points in the space can create how many planes?

a)10

b)8

c)5

e)4

One plane can be created with at least three points. So C_5^3. The final answer is 10. But aren't there planes which are passing through 4 or 5 points? Or they will again be repeated?

 

[Defennder]

You're asking for the maximum possible number of planes that can be formed by 5 points, not unlikely possibility that all 5 points may reside on the same plane. That's pretty much the same thing as the first question above.

 

[Physicsissuef]

I think more than 10 will be created without any repetition.

 

[symbolipoint]

The question reminds me of the Handshake problem. Read about THAT one. If you understand the solution methods, then this 5 Points problem is no more difficult than the Handshake problem.

 

[HallsofIvy]

Yes. Ok, thank you. And what about this one:

Five points in the space can create how many planes?

a)10

b)8

c)5

e)4

One plane can be created with at least three points. So C_5^3. The final answer is 10. But aren't there planes which are passing through 4 or 5 points? Or they will again be repeated?

Only if the "4 or 5 points" happen to lie on a single plane. In that case there would be fewer planes. That's a very special situation. You are clearly expected to find the maximum number of planes which would mean the points are in "general position". I.e. not 4 or more points lie on the same plane.