Question

State whether the following statement is true or false.
There are an infinite number of lines that pass through two distinct points.
A. TRUE
B. FALSE

Answer 1

Hint: We will find this statement is true or false using the Euclid’s axiom which states that given two distinct points, there is a unique line that passes through them.

Complete step-by-step answer:
We know Euclid’s axiom has the statement “Given two distinct points, there is a unique line that passes through them.”
We take two points A and B anywhere on the surface which are distinct from each other.

     

Therefore by Euclid’s axiom stated above we can say that a unique line passes through two distinct points.
Therefore, the statement given in the question is FALSE.

So, the correct answer is “Option B”.

Note: Alternate method:
We take two points A and B anywhere on the surface which are distinct from each other.
If we consider two points separately we can say
Point A has an infinite number of lines passing through it.
Similarly, point B has an infinite number of lines passing through it.

     

But when we take both points A and B together, we fix two points through which the line should pass, which means there should be at least one line passing through two distinct points.
The line m goes beyond the two fixed points, but when the two points are fixed then we write line segment AB instead of line m.
Now, if we try to draw any other line joining the two points it will always overlap the unique line.
So, we can say there is a unique line which passes through two distinct points A and B.
Therefore, the statement is FALSE.