MAPPINGS AND TRANSFORMATIONS
A.........
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215
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A glide reflection Gz,AB is simply a translation TAB followed by a reflection Rz
about a line l that is parallel to AB. We will prove that this is the only other additional
isometry later.
It is obvious that all isometries have inverses and that the inverses themselves must
also be isometries. Not quite so obvious is the fact that an isometry also preserves
straight lines.
Theorem 7.2.3. (lsometries Preserve Straight Lines)
(I) Let P, Q, and R be three points, and let P', Q', and R' be their images under
an isometry. The points P, Q, and R are collinear, with Q between P and R,
if and only if the points P', Q', and R' are collinear, with Q' between P' and
R'.
(2) Let l be a straight line, and let l' be the image of l under an isometry. Then l'
is a straight line.
Proof. Here we write lAB I for dist(A, B).
(1) We will show that if Q is between P and R, then Q' must be between P' and
R' (the proof of the converse may be obtained by interchanging P, Q, and R
with P', Q', and R').
R
p
....--
P'
_____ .... -------
----------Q·:---------;;.
216
THE EUCLIDEAN TRANSFORMATIONS OR ISOMETRIES
If Q is between P and R, then
IPQI + IQRI = IPRI.
Since an isometry preserves distances, we must have
IP'Q'I = IPQI,
Hence,
IQ'R'I = IQRI,
and
IP'R'I = IPRI.
IP'Q'I + IQ'R'I = IP'R'I,
and the Triangle Inequality shows that P', Q', and R' are collinear with Q'
between P' and R'.
(2) Let P and Q be two points on l, and let P' and Q' be their images under an
isometry. Let m be the line passing through P' and Q'. We will show that m
is the image of l under the isometry. We must check two things:
(a) that every point R on l has its image R' on m and
(b) that every pointS' on m has its preimage Son l.
It follows from statement (1) above that if R is a point on l other than P or Q,
then P', Q', and R' must be collinear, so R' is a point on m. Conversely, if S'
is on m, then P', Q', and S' are collinear, and it follows again from statement
(1) that P, Q, and S are on l.
D
The next theorem tells us that an isometry preserves the shape and size of all of the
geometric figures.
Theorem 7.2.4. Under an isometry,
( 1) the image of a triangle is a congruent triangle;
(2) the image of an angle is a congruent angle;
(3) the image of a polygon is a congruent polygon;
(4) the image of a circle is a congruent circle.
Proof. We will prove statement ( 1) and leave the rest as exercises. Let P, Q, and R
be the vertices of a triangle. It follows from Theorem 7 .2.3 that their images P', Q',
and R' are the vertices of a triangle and that the edges
P'Q',
Q'R',
and
P'R'