Question

congruence transformations
another name for a rigid motion or a combination of rigid motions is
a congruence transformation because the preimage and image are congruent.
the terms rigid motion and congruence transformation are interchangeable.
example 2 describing a congruence transformation
describe a congruence transformation that maps \\( \square a b c d \\) to \\( \square e f g h \\).
solution
two sides of \\( \square a b c d \\) rise from left to right,
and the corresponding sides of \\( \square e f g h \\) fall
from left to right. if you reflect \\( \square a b c d \\)
in the \\( y \\)-axis, as shown, then the image,
\\( \square a ^ { \prime } b ^ { \prime } c ^ { \prime } d ^ { \prime } \\), will have the same orientation
as \\( \square e f g h \\).
then you can map \\( \square a ^ { \prime } b ^ { \prime } c ^ { \prime } d ^ { \prime } \\) to \\( \square e f g h \\)
using a translation 4 units down.
so, a congruence transformation that maps \\( \square a b c d \\) to \\( \square e f g h \\) is a reflection
in the \\( y \\)-axis, followed by a translation 4 units down.
check verify that the corresponding sides and the corresponding angles are congruent.
\\( a b = e f = 2 \\) and \\( c d = g h = 2 \\), so \\( \overline { a b } \cong \overline { e f } \\) and \\( \overline { c d } \cong \overline { g h } \\).
using the distance formula, \\( b c = f g = \sqrt { 5 } \\) and \\( a d = e h = \sqrt { 5 } \\), so
\\( \overline { b c } \cong \overline { f g } \\) and \\( \overline { a d } \cong \overline { e h } \\).
using a protractor to estimate the angle measures, you can see that the
corresponding angles appear to be congruent. \\( \checkmark \\)
self-assessment

1. reasoning a composition of rigid motions maps one figure to another figure. is the image at each step of the

composition congruent to the preimage and the final image? explain.

1. in example 2, describe another congruence transformation that maps \\( \square a b c d \\) to \\( \square e f g h \\).
2. describe a congruence transformation that maps \\( \triangle j k l \\) to \\( \triangle m n p \\).

Explanation:

Step1: Analyze the position of \( \triangle JKL \) and \( \triangle MNP \)

First, observe the orientation and position of the two triangles. Notice that \( \triangle JKL \) can be rotated \( 180^{\circ} \) around the origin.

Step2: Determine the translation after rotation

After a \( 180^{\circ} \) rotation around the origin, the rotated triangle needs to be translated. Count the units of translation.

Answer:

A congruence transformation that maps \( \triangle JKL \) to \( \triangle MNP \) is a \( 180^{\circ} \) rotation about the origin followed by a translation \( 1 \) unit right.