Consider the transformation T on the coordinate plane where

T:(x,y)→(2x+y,x+2y).

Prove the the images of any two circles of radius 1 are congruent under T.

Find the area enclosed by the image of the unit circle under T.

Very interesting, Jer. Here's my take:

Part 1)

Consider a line segment has that goes from (a,b) to (a+c,b+d).

Before transformation, it has a slope of d/c and a length = sqrt(c^2 + d^2)

After transformation, it goes from (2a+b,a+2b) to (2a+2c+b+d, a+c+2b+2d).

Its slope after transformation is (2d+c)/(2c+d).

Its length after transformation is sqrt((2d+c)^2 + (2c+d)^2).

The salient point is that the slope and length after the transformation does not depend on a or b. Two line segments with identical slope and length before the transformation will have identical slopes and lengths afterwards.

Note two implications of this:

a) The transformed midpoint of a line is the midpoint of the transformed line. And by extension, this means that transformation (T) preserves relative lengths on a line seqment.

b) Two shapes which are congruent and identically oriented with respect to the x-y axes will still be congruent after the transformation. Since a circle only has one orientation with respect to the x-y plane, this means that the transformation of two congruent circles will still be congruent.

Part 2)

Consider a unit square oriented with its sides parallel to the x and y axes. In particular, consider the square whose corners are at (0,0), (0,1), (1,0) and (1,1). After transformation it has corners at (0,0), (1,2), (2,1) and (3,3). It is now a diamond, and with little work we can see that it has an area of 3. In other words, the transformation (T) has tripled its area.

Because the T preserves relative lengths, this means that T will triple the area of any square oriented with its sides parallel to the x and y axes.

And because any closed shape on the x-y plane can be formed as the limit of tiled squares all of whose sides are parallel to the x and y axes, this means that T will triple the area of any closed shape on the x-y plane.

In particular, T will change the area of a unit circle from Pi to 3*Pi.