Prove that the transformation F R2 rightarrow R2 is an isome

Prove that the transformation F: R^2 rightarrow R^2 is an isometry in Euclidean geometry, where F(x, y)= (-x + 2. -y + 1).

Solution

Let P=(x,y) and Q=(u,v) be two points in R² .

To show that d(P,Q) = d(F(P),F(Q)), ...............................(1)

.where d is the Euclidean distance.(An isometry is a transformation that preserves distances).

Equivalently, we will show

d(P,Q) ²= d(F(P),F(Q))²......................(2)

LHS = (x-u)² + (y-v)² ...............................(3)

RHS = (-x+2-(-u+2))² + (-y+1-(-v+1))²

          = (-x+u)² + (-y+v)²

         =(x-u)² + (y-v)² ..............................(4)

Thus F is an isometry