GEOMETRYHELP

DO NOW

What is an isometry?

What is a rigid motion?

GEOMETRYHELP

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

Compositions of ReflectionsLESSON 9-6

Additional Examples

GEOMETRYHELP

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.

Compositions of ReflectionsLESSON 9-6

Additional Examples

Quick Check

GEOMETRYHELP

GEOMETRYHELP

GEOMETRYHELP

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.

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.

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

Compositions of ReflectionsLESSON 9-6

Additional Examples

Quick Check

GEOMETRYHELP

GEOMETRYHELP

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

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.

Compositions of ReflectionsLESSON 9-6

Additional Examples

GEOMETRYHELP

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

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

(continued)

Compositions of ReflectionsLESSON 9-6

Additional Examples

Quick Check

GEOMETRYHELP

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 isa rotation of N.

The segment connecting the vertical segments of the letter N slopesdown from the left to the right. The segment connecting the verticalsegments of the figure on the right slopes up from left to right. So thefigure on the right and N have opposite orientations. It is a reflectionof N across a horizontal line.

Compositions of ReflectionsLESSON 9-6

Additional Examples

Quick Check