# 9 6 Compositions of Reflections

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"9 6 Compositions of Reflections"

```					Compositions of Reflections
LESSON 9-6

Judging by appearances, is one figure a translation image or
rotation image of the other? Explain.

The figures appear to be congruent, and their orientations are the same.

Corresponding sides of the figures appear to be parallel.

This suggests that one figure is a translation image of the other and not a
rotation image.
Quick Check

HELP                                                                GEOMETRY
Compositions of Reflections
LESSON 9-6

Find the image of the figure for a reflection across line   and
then across line m.

First, find the reflection image in line .
It no longer looks like a 4.

Then, find the image of the first reflection in line m. The final image is a
translation of the original figure.
The arrow shows the direction and distance of the translation.

The arrow is perpendicular to lines and m with
length equal to twice the distance from to m.
Quick Check

HELP                                                                     GEOMETRY
Compositions of Reflections
LESSON 9-6

The letter D is reflected across line x and then across line y.
Describe the resulting rotation.

Find the image of D through a reflection in line x.

Find the image of the reflection through
another reflection in line y.

The composition of two reflections in intersecting lines is a rotation. The
center of rotation is the point where the lines intersect, and the angle is
twice the angle formed by the intersecting lines.
So the letter D is rotated 86° clockwise, or 274° counterclockwise, with the
center of rotation at point A.                                     Quick Check

HELP                                                                GEOMETRY
Compositions of Reflections
LESSON 9-6

ABC has vertices A(–4, 5), B(6, 2), and C(0, 0). Find the
image of ABC for a glide reflection where the translation is
(x, y) (x, y + 2) and the reflection line is x = 1.

First, translate ABC by the rule
(x, y) (x, y + 2).

(–4, 5)    (–4 + 0, 5 + 2), or (–4, 7)

(6, 2)    (6 + 0, 2 + 2), or (6, 4)

(0, 0)    (0 + 0, 0 + 2), or (0, 2)

HELP                                                                   GEOMETRY
Compositions of Reflections
LESSON 9-6

(continued)

Then, reflect the translated image across the
line x = 1.

The glide reflection image         A B C has vertices A (6, 7), B (–4, 4), and
C (2, 2).

Quick Check

HELP                                                                   GEOMETRY
Compositions of Reflections
LESSON 9-6

Tell whether orientations are the same or opposite. Then
classify the isometry of the letter N.

If you turn the figure on the left, it has the same orientation as N. It is
a rotation of N.

The segment connecting the vertical segments of the letter N slopes
down from the left to the right. The segment connecting the vertical
segments of the figure on the right slopes up from left to right. So the
figure on the right and N have opposite orientations. It is a reflection
of N across a horizontal line.
Quick Check

HELP                                                                  GEOMETRY

```
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