Solution to Exercise 3.37.
Happy is the man who can understand why things happen.
—Virgil (70–19 BCE)

Exercise 3.37. Let A(0, 0, 1), B(1, 0, 1), C(0, 1, 1), D(1, 1, 1), E(2, 1, 1), and F(1, 2, 1). Show the sets {A, B, C} and {D, E, F} are congruent. (Two sets of points are said to be congruent provided there is an isometry where one set is the image of the other set.)  

Find the matrix M of an isometry. Here, we find a direct isometry.

 

Then the images of the points are

 

Hence, a = b = 1.

 

Hence,  and  We check with the last pair of points:

 

Therefore,

 

is the matrix of an isometry that maps {A, B, C} to {D, E, F}. Hence, {A, B, C} is congruent to {D, E, F}.//