Location of new point in new rectangle I am stuck on a problem that maybe trivial but I am stumped.  Suppose that there is a rectangle with points $(5,0), (20,0), (5,5) \text{and} (20,5)$. Inside the rectangle, there is a point $(6,1)$. Doing modifications to the rectangle, we get a new rectangle with points $(10,1),(16,1),(10,4),(16,4)$. I was wondering were the point related to $(6,1)$ would be located? I was also wondering if this process of finding these new points can be generalized as well?
Thank you
 A: One natural transformation that maps the first rectangle to the second one is:
$$
x' = (x-5)\cdot \frac{16-10}{20-5}+10,
\quad
y' = (y-0)\cdot \frac{4-1}{5-0}+1
$$
The first expression maps the interval $[5,20]$ to the interval $[10,16]$.
The second expression maps the interval $[0,5]$ to the interval $[1,4]$.
A: The transformation includes a different scaling of the rectangle dimensions. Because it passes from a $15\times 5$  to a $6\times 3$ proportion, the width (measured along the $x$- coordinates) is scaled by $2/5$, the height (measured along the $y$- coordinates) by $3/5$.
Now let us consider the distances of the point $(6,1)$ from the lower left corner of the first rectangle, i.e., the point $(5,0)$. The distance is $1$ for both the $x$ and $y$ coordinates. These distances must be scaled by $2/5$ and $3/5$, respectively. That is to say, they are $2/5$ and $3/5$, respectively.
Taking into account not only the scaling, but also the translation,  with the lower left corner of the new rectangle in $(10,1)$, we get the new coordinates $(10.4,1.6)$.
