Does there exists two different curves from $ \ (0,0) \ \ to \ \ (10,0) \ $ having same arc length?

Question

Does there exists two different curves from $ \ (0,0) \ \ to \ \ (10,0) \ $ having same arc length ?

Answer:

I think there does not exists such two different curves.

For, one of the curve is the straight line segment $ \ x=10t , \ 0 \leq t \leq 1 \ $

This line segment has curve length $ \ =10 \ $

Any other curve joining $ (0,0) \ \ to \ \ (10,0) \ $ must be different with different arc length.

Help me with better way.

Answer 1

Guide:

Try to construct a circle with centered at $(5,0)$, making it passes through $(0,0)$, and $(10,0)$.

Answer 2

Well yes, there are: Take an arbitrary curve $l$ which is not a straight line. Then take the straight line between the points and reflect $l$. This way you get a second curve having the same arc length by symmetry.

Answer 3

You have asserted that there are not two different curves with the same arc length, and your proof is that no curve has the same length as one particular curve you have given. This is not a proof, any more than "15 is prime, because 2 doesn't divide it" is.