Problem 254

Beginning with the point $(5,11)$, and applying $R$ then $T$, the resulting coordinates are $(11,-3)$, as shown: $$(5,11) \overset{R}{\longrightarrow} (11,-5) \overset{T}{\longrightarrow} (11,-3)$$
Solution 1:

When a point is rotated $90\degree$ about the origin $4$ times, the result is a rotation of
$4\times90\degree=360\degree$ or one
full rotation about the origin.

Thus beginning with the point $(-3,7)$, when $R$ is applied $4$ times, the point returns to its
original location, and so the resulting coordinates are $(-3,7)$.

When $R$ is applied a 5th time, the
resulting coordinates are $(7,3)$.

Solution 2:

Beginning with the point $(-3,7)$,
and applying $R$ $5$ times, the resulting coordinates are
$(7,3)$, as shown: $$(-3,7) \overset{R}{\longrightarrow} (7,3) \overset{R}{\longrightarrow} (3,-7)\overset{R}{\longrightarrow} (-7,-3)\overset{R}{\longrightarrow} (-3,7)\overset{R}{\longrightarrow} (7,3)$$
Beginning with the point $(9,1)$, and applying the sequence $R$, $R$, $T$, the resulting coordinates are $(-9,1)$, as shown: $$(9,1) \overset{R}{\longrightarrow} (1,-9) \overset{R}{\longrightarrow} (-9,-1) \overset{T}{\longrightarrow} (-9,1)$$ Continuing with the point
$(-9,1)$, and applying the sequence
$R$, $R$, $T$
again, the resulting coordinates are $(9,1)$, as shown: $$(-9,1) \overset{R}{\longrightarrow} (1,9) \overset{R}{\longrightarrow} (9,-1) \overset{T}{\longrightarrow} (9,1)$$ Beginning with the point $(9,1)$, and applying the sequence $R$, $R$, $T$
twice, the resulting coordinates are $(9,1)$ (that is, the point returns to its
original location).

This will continue to occur each time the sequence $R$, $R$, $T$
is applied an even number of times, and so after applying $R$, $R$, $T$
$10$ times, the resulting coordinates
are $(9,1)$.

Beginning with the point $(9,1)$, and
applying the sequence $R$, $R$, $T$
an 11th time, the resulting coordinates are $(-9,1)$, the steps to which were
previously shown.