5 points (last one i swear)

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In summary, the conversation discusses the existence of a line segment containing another lattice point between two given lattice points in the plane. It is proven using the Pigeonhole Principle and the midpoint formula. The case of parity is considered, with 4 possibilities depending on the parity of the points.
  • #1
barbiemathgurl
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im so embarrased askin so much :blushing:

show that given 5 distinct lattice points in the plane (points with integer coordinates) there exists a line segment between both of them containing another lattice point on its interior.
 
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  • #2
Consider parity.
In which case there are 4 possibilities:
(even,even);(even,odd);(odd,even);(odd,odd)

Now there are 5 points so by the Pigeonhole Principle two of them are of equal parity. Say (a,b) and (c,d).

By the midpoint formula we have that [(a+c)/2,(b+d)/2] is a midpoint of the line joining them. But since "a" and "c" have similar parity it means (a+c)/2 is an integer. Likewise, (b+d)/2 is an integer. That shows that this point lies on the interior of this line segment.
 
  • #3


Don't be embarrassed for asking questions and seeking help! It's always better to clarify any doubts and learn something new. And in mathematics, asking questions is a crucial part of the learning process.

Now, for the problem at hand, let's first define what a lattice point is. A lattice point is a point with integer coordinates, meaning both the x and y coordinates are whole numbers. So, given 5 distinct lattice points, we can label them as (x1, y1), (x2, y2), (x3, y3), (x4, y4), and (x5, y5).

Now, let's consider the line segment connecting (x1, y1) and (x2, y2). Since both points have integer coordinates, we can express this line segment as y = mx + b, where m and b are also integers. This line segment will pass through an infinite number of lattice points.

Next, let's consider the line segment connecting (x2, y2) and (x3, y3). Similarly, this line segment can also be expressed as y = mx + b, where m and b are integers. And since this line segment also passes through an infinite number of lattice points, it is bound to intersect the previous line segment at some point, creating a new lattice point on its interior.

We can repeat this process for the remaining line segments, (x3, y3) to (x4, y4) and (x4, y4) to (x5, y5), always finding a new lattice point on the interior.

Therefore, we have shown that given 5 distinct lattice points, there exists a line segment between both of them containing another lattice point on its interior. This is a property of lattice points and lines in the plane, and it holds true for any number of lattice points. So, don't worry, you're not asking too much! Keep asking questions and keep learning.
 

1. What are the 5 points?

The 5 points refer to the five key components or factors that are being considered or discussed in a particular situation or topic.

2. What is the significance of the last point?

The last point may hold the most weight or importance in the overall discussion or conclusion.

3. Can you give an example of a situation where the 5 points are used?

One example is when evaluating a scientific experiment, where the 5 points may include the hypothesis, materials, methods, results, and conclusion.

4. How do the 5 points relate to the scientific method?

The 5 points are closely related to the scientific method as they help to structure and guide the process of conducting and analyzing scientific research.

5. Are the 5 points always the same or can they vary?

The 5 points can vary depending on the context and purpose of the discussion or research. However, they often encompass the key elements that are necessary for a thorough analysis or evaluation.

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