Non-Isometric of the Wolpert's problem

Question

Related to the document Drums That Sound the Same of S. J. Chapman, he has proved those two shapes

was isopectrals.

Question : I don't know how to prove that they are not isometric (Riemannian isometry), and I didn't find out any references on the subject. Could anyone be able to prove in details that both shapes are not isometric?

Thanks!

Answer 1

The two vertices with acute angles are at distance $\sqrt{10}$ in the first, $\sqrt{2}$ in the second.

EDIT:

There are many other differences. For example, the longest edge of the first polygon has length $2$, of the second $2 \sqrt{2}$.