Known:
    The vertices: a.x, a.y, b.x, b.y, c.x, c.y, d.x, d.y

Unknown:
    the scaling factor: k
    the rotation factor: theta
    the translation vector: v.x, v.y

matrix form:

    [scaling matrix] * [rotation matrix] * [translation matrix] * [a and b] == [c and d]

    +--       --+     +--                          --+     +--         --+     +--       --+    +--       --+
    | k   0   0 |     | cos(theta)   -sin(theta)   0 |     | 1   0   v.x |     | a.x   b.x |    | c.x   d.x |
    | 0   k   0 |  *  | sin(theta)   cos(theta)    0 |  *  | 0   1   v.y |  *  | a.y   b.y | == | c.y   d.y |
    | 0   0   1 |     | 0            0             1 |     | 0   0   1   |     | 1     1   |    | 1     1   |
    +--       --+     +--                          --+     +--         --+     +--       --+    +--       --+

    multiply first three matrices

    +--                                                                                    --+     +--       --+    +--       --+
    | k * cos(theta)      k * -sin(theta)     (k * cos(theta) * v.x + k * -sin(theta) * v.y) |     | a.x   b.x |    | c.x   d.x |
    | k * sin(theta)      k * cos(theta)      (k * sin(theta) * v.x + k * cos(theta) * v.y)  |  *  | a.y   b.y | == | c.y   d.y |
    | 0                   0                   1                                              |     | 1     1   |    | 1     1   |
    +--                                                                                    --+     +--       --+    +--       --+

Equivalent series of equations:
    (k * cos(theta) * a.x + k * -sin(theta) * a.y + (k * cos(theta) * v.x + k * -sin(theta) * v.y)) = c.x
    (k * cos(theta) * b.x + k * -sin(theta) * b.y + (k * cos(theta) * v.x + k * -sin(theta) * v.y)) = d.x
    (k * sin(theta) * a.x + k * cos(theta) * a.y + (k * sin(theta) * v.x + k * cos(theta) * v.y)) = c.y
    (k * sin(theta) * b.x + k * cos(theta) * b.y + (k * sin(theta) * v.x + k * cos(theta) * v.y)) = d.y

Goal:
    solve for k, theta, and v