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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
In the diagram, is reflected in line r, and is reflected in line s.
,r s CD' 'C D
8. Which lines are perpendicular to ''?DDsuuur
r and s
In the diagram, is reflected in line r, and is reflected in line s.
,r s CD' 'C D
9. Name a segment parallel to ''.CCsuuur
''DD
In the diagram, is reflected in line r, and is reflected in line s.
,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
In the diagram, is reflected in line r, and is reflected in line s.
,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
Identify the number of line and rotational symmetry of the figure shown.
Line: _____________
Rotational: _________
none
none
Identify the number of line and rotational symmetry of the figure shown.
3
120°
360 3
= 120°
Line: _____________
Rotational: _________
90°, 180°
360 4
= 90°
Identify the number of line and rotational symmetry of the figure shown.
Line: _____________
Rotational: _________
4
60°, 120°,
360 6
= 60°
Identify the number of line and rotational symmetry of the figure shown.
Line: _____________
Rotational: ____________
6
180°
Identify the number of line and rotational symmetry of the figure shown.
Line: _____________
Rotational: _________
1
none
Identify the number of line and rotational symmetry of the figure shown.
4
90°, 180°
360 4
= 90°
Line: _____________
Rotational: _________
Identify the number of line and rotational symmetry of the figure shown.
1
none
Line: _____________
Rotational: _________
Identify the number of line and rotational symmetry of the figure shown.
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)