Download - 9.5 & 9.6 – Compositions of Transformations & Symmetry.

9.5 & 9.6 – Compositions of Transformations & Symmetry

Glide Reflection:

A transformation with a translation and then a reflection in a line parallel to the direction of the translation

PQ

'P

'Qk

''P

''Q

Composition of Transformations:

Any two transformations combined to form a single transformation

If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a ___________. If ____ is the image of P after the two reflections, then:translation ''P

• '' is perpendicular to and PP k m

• '' 2 , where is the distance between

and

PP d d

k m

=

If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a _________. If ____ is the image of P after the two reflections, then:

rotation A ''

• The angle of rotation is 2x°, where x° is the measure of the acute or right angle formed by k and m

Graph the image of A(1, –2) after the described glide reflection.

1. Translation: (x, y) → (x + 2, y) Reflection: in the x-axis

A 'A

''A

Graph the image of A(1, –2) after the described glide reflection.

2. Translation: (x, y) → (x – 1, y + 3) Reflection: in x = 2

A

'A ''A

The vertices of ABC are A(3, 1), B(1, 5) and C(5, 3). Graph the image of ABC after a composition of the transformations in the order they are listed.

A

B

C

'A

Reflection: y-axisTranslation: (x, y) → (x + 3, y – 5)

'C

'B

''A

''C

''B

The vertices of ABC are A(3, 1), B(1, 5) and C(5, 3). Graph the image of ABC after a composition of the transformations in the order they are listed.

Reflection: y = 1Reflection: y = x

A

B

C

(y, x)(x, y) →

(3, 1)

(1, 3)

'A =

''A =

(1, –3)

(–3, 1)

'B =

''B =

(5, –1)

(–1, 5)

'C =

''C =

'C

'B

'A=

''A

''C

''B

Describe the composition of transformations.

Reflection: x-axisTranslation: (x, y) → (x + 6, y + 2)

Describe the composition of transformations.

Rotation: 90°Reflection: x-axis

In the diagram, is reflected in line r, and is reflected in line s.

,r s CD' 'C D

7. A translation maps onto which segment?CD

'' ''C D


,r s CD' 'C D

8. Which lines are perpendicular to ''?DDsuuur

r and s


,r s CD' 'C D

9. Name a segment parallel to ''.CCsuuur

''DD


,r s CD' 'C D

10. If the distance between r and s is 2.4 inches, what is the length of ''?CC

2.4 2

4.8 inches


,r s CD' 'C D

11. Is the distance from to r the same as the distance from C to r?

'C

Yes,

Def. of Reflection

12. Find the angle of rotation that maps T onto ''T

75 2

150°

13. Find the angle of rotation that maps A onto ''A

99 2

198°

Line of Symmetry:

The figure can be mapped onto itself by a reflection in the line

Rotational Symmetry:

The figure can be mapped onto itself by a rotation of 180° or less around a point

Identify the number of line and rotational symmetry of the figure shown.

Line: _____________

Rotational: _________

1

none


Line: _____________


none

none


3

120°

360 3

= 120°

Line: _____________


90°, 180°

360 4

= 90°


Line: _____________


4

60°, 120°,

360 6

= 60°


Line: _____________

Rotational: ____________

6

180°


Line: _____________


1

none


4

90°, 180°

360 4

= 90°

Line: _____________



1

none

Line: _____________



2Line: _____________

Rotational: _________180°

360 2

= 180°

HW Problems

9.5 #13

9.5

9.6

611-612

621-623

3, 6, 9, 11, 13-21 all

3, 4, 6, 7, 9, 11, 12

Translation:

Rotation: 180°

(x, y) → (x + 5, y + 1)