advanced geometry: horizontal translation is an isometry

[stevew2344]

Show that the horizontal translation Φ(x,y) = (x + a, y) for any fixed real number a is an isometry in the hyperbolic plane.

dont know where to start on this one!

 

[daon2]

Use your definition of an isometry, and your notion of distance. You show \(\displaystyle d(P,Q) = d(\Phi(P), \Phi(Q))\), where \(\displaystyle P,Q\) are points in the upper-half plane.