This is in continuation of my earlier post here.
On pg.#6-7, question #5 is about uniform growth used to form one rectangle from another by shrinking or expanding $x,y$ coordinates. The different parts of the question are attempted below with need for inputs in part (b), & vetting for the rest:
a) Describe all the points in the lh-plane that represent the
rectangles uniformly grown from the rectangle $(3, 2)$.
-> $(3\pm i, 2\pm i), i\in \mathbb{R}$
b) Given the rectangle $(3, 2)$ and the rectangle $(5, 3)$, can you find a rectangle that can be uniformly grown from each of them? Explain.
-> Incomplete attempt - Need take differences in $x,y$ coordinates of the two points, and form a line joining the two, with slope =
c) If one rectangle is uniformly grown from another, can
they be similar? Describe all such pairs of rectangles.
-> Yes, the answer is $(a\pm i, b\pm i), i\in \mathbb{R}$, with $(a,b)$ the start point of the rectangle.
d) Is it possible to grow (or shrink) uniformly a square from
any rectangle? Explain.
-> No, except rectangle not possible to generate for 'possibly' one instance where the two points become equal. Otherwise, if the start rectangle is a square; then always possible.
Although, am still to prove that from a rectangle there can be at max. one instance where square can be produced. Request idea for that apart from part (b).
Guide:
If they grow at the same rate, then they can be written as $(3+t, 2+t)$ for some $t\ge0$. The same parameters are used in both entry of the coordinate so that the growth is uniform.
To answer $(b)$, try to solve whether you can find a solution for $(3+s, 2+s)=(5+t,3+t)$.
For part $(c)$, the question is equivalent to consider rectangle being characterized by $(a,b)$ and the growth rectangle is $(a+t, b+t)$, is it possible that they are similar? if so, what are the characterization of $a$ and $b$.
For part $(d)$, You might like to fix a particular rectangle and show that you can't grow a square.