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\textbf{\large MAT 515 Homework 9 - Fall 2016}
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\begin{enumerate}
\item An isometry maps segment to congruent segments, angles to congruent angles, circles to congruent circles.
\item A reflection about a line is an isometry.
\item A symmetry about a point is an isometry. (Can you prove it without using the fact that
a symmetry about a point is a special kind of rotation?)
\item A rotation about a point is an isometry.
\item Each point in the plane is uniquely determined by its distance to three non-collinear
points. (Thus, for every triple of non collinear points $A,B,C$, if $P$ and $P_1$ are points, and $PA = P_1A$, $PB = P_1B$ and $PC = P_1C$ then
$P = P_1$)
\item The composition of two translations is translation and that for each triple of points A, B, C in the plane, $T_{BC} \circ T_{AB} = T_{AC}$ . Also, two translations commute, that is, for
each four points in the plane, $T_{AB }\circ T_{CD} = T_{CD} \circ T_{AB}.$
\item Does the composition of two symmetries about points commute? What about two re-
flections about two lines? Two rotations? (Recall that two transformations S and T
\item A translation maps a line segment to a parallel line segment.
\item If $\alpha$ is an angle different from 0, then the only fixed point of a rotation about point P through $\alpha$ is P.
\item The set of fixed points of a reflection about a line is the line.
\item Prove that the composition of $h_A \circ h_A$ of half-turn about a point A with itself is the
identity. Deduce that the inverse of a half-turn about a point is the same half-turn.
about the same point. Formulate and prove analogous statements about reflections.
\item Prove that a function from the plane to the plane that preserve distances is bijective.
\item Find all the symmetries of the following figures: (that is, all the isometries that leave each of the figures invariant. A figure is invariant under an isometry if the image of the
figure by the isometry coincides with the figure, not necessarily pointwise.)
\begin{enumerate}
\item An isosceles triangle.
\item An equilateral triangle.
\item A rectangle.
\item A square.
\item A circle.
\item A regular n-sided polygon.
\end{enumerate}
\end{enumerate}
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\item We studied five types of isometries: translations, rotations, symmetries about a point, reflections about a line and glide reflections. Make a table as we did in class, studying what happens when one compose two isometries, one of each of the five types. You need to consider twenty five cases, and some have sub cases!