Question

Show that a dilation σ by a factor of r scales all distances by a factor of r. That is, for all points A and B, we have:

|σ(A)σ(B)| = r|AB|

Answer 1

Let us solve the statement using a simple example.

Let us consider the dilation of a Triangle ABC by a factor of 2.

The points taken in solution can be referred from the figure given below.

Notice that every coordinate is multiplied by a scale factor, r of 2.

now lets consider two points on ABC. Say, P(1.5,0) and Q(-1,0)

distance between these points =

                                                               =

                                                               =

                                                               =

Now ,i.e. dilation of P(1.5,0) will be P'(3,0)

and ,i.e. dilation of Q(-1,0) will be Q'(-2,0)

distance between P' and Q', =

                                                             =

                                                             =

Which implies that ,   

Hence A dilation by r factor scales all distances by a factor of r.