Prove that only one straight line passes through two point

In summary, to prove that only one straight line passes through two points, one can use the slope-intercept form of a line and check if the slopes and y-intercepts are the same for both points. This is important because it establishes a unique relationship between the points and has various applications. It is not possible for there to be more than one straight line passing through two points, even if the points have the same coordinates, as this would result in an undefined slope. This rule has no exceptions and is a fundamental principle in mathematics.
  • #1
parshyaa
307
19
I was just thinking of basic definitions of geometry and i came to this question, so how could i prove that only one straight line passes through two distinct points.
 
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  • #2
You have to require that in the axioms, otherwise you get non-euclidean geometry where multiple different lines can go through two points.
 
  • #3
Simple example - non-Euclidean: sphere, where all longitude lines go through both poles.
 

1. How do you prove that only one straight line passes through two points?

To prove that only one straight line passes through two points, you can use the slope-intercept form of a line: y = mx + b. Plug in the coordinates of the two points to get two equations. If the slopes (m) of the two equations are the same and the y-intercepts (b) are different, then you have proven that only one straight line can pass through those two points.

2. Why is it important to prove that only one straight line passes through two points?

It is important to prove that only one straight line passes through two points because it helps establish a unique relationship between those two points. This can be used in various mathematical and scientific applications, such as geometry, graphing, and calculating distances.

3. Can there be more than one straight line passing through two points?

No, there can only be one straight line passing through two points. This is because two points determine a unique line, and any additional points would either lie on that same line or create a different line altogether.

4. What if the two points have the same coordinates?

If the two points have the same coordinates, it means they are the same point and therefore cannot have a line passing through them. This would result in an undefined slope and the equation y = mx + b would not hold true.

5. Are there any exceptions to this rule?

No, there are no exceptions to this rule. The concept that only one straight line can pass through two points is a fundamental principle in mathematics and cannot be disproven.

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