Math 5270
Transformational Geometry
Day 5
Summer 13
Day 5, Summer 13
Math 5270 Transformational Geometry
1/19
Composition of rotations
Composing rotations about the same point yields a rotation.
Such rotations commute.
Day 5, Summer 13
Math 5270 Transformational Geometry
2/19
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Composition
of arotations
Composing rotations about the same point yields a rotation.
Such rotations commute.
Composing rotations about different points yield a rotation...
Day 5, Summer 13
Math 5270 Transformational Geometry
2/19
Produced
with a Trial
Use
your geogebra
to Version of PDF Annotator - www.PDFAnno
Make an angle slider for angle α between 0◦ and 360◦
In input bar write β = 360◦ − α
Pick any two points A and B and observe the composition of
rotation about A by α, followed by rotation about B by β.
Day 5, Summer 13
Math 5270 Transformational Geometry
3/19
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with a Trial
Group
of rotations
R?Version of PDF Annotator - www.PDFAnno
Day 5, Summer 13
Math 5270 Transformational Geometry
4/19
Produced
Trial Version
PDF Annotator
- www.PDFAnno
Equation
forwith
thea rotation
not ofaround
the origin?
Day 5, Summer 13
Math 5270 Transformational Geometry
5/19
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Alternative?
Day 5, Summer 13
Math 5270 Transformational Geometry
6/19
Produced
Trialisometries
Version of PDF
Annotator
How
can wewith
tella two
are the
same?- www.PDFAnno
Day 5, Summer 13
Math 5270 Transformational Geometry
7/19
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Reflections
Definition
wthout coordinates
with coordinates
Day 5, Summer 13
Math 5270 Transformational Geometry
8/19
Reflections
Definition
wthout coordinates
with coordinates
Hmm... I think I might need more space...
Day 5, Summer 13
Math 5270 Transformational Geometry
8/19
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a Trial Version of PDF Annotator - www.PDFAnno
Equation
forwith
a reflection
Day 5, Summer 13
Math 5270 Transformational Geometry
9/19
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Produced
a Trial Version
Equation
forwith
a reflection,
takeof2PDF Annotator - www.PDFAnno
Any super easy reflections you know?
Day 5, Summer 13
Math 5270 Transformational Geometry
10/19
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with
a Trial
Version of PDF Annotator - www.PDFAnno
Fixed
points,
fixed
sets?
Day 5, Summer 13
Math 5270 Transformational Geometry
11/19
Produced
with when
a Trialyou
Version
of PDFtwo
Annotator
- www.PDFAnno
What
happens
compose
reflections?
Use your geogebra to observe what happens when you compose
different reflections.
Why did I say different?
Day 5, Summer 13
Math 5270 Transformational Geometry
12/19
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Day 5, Summer 13
Math 5270 Transformational Geometry
13/19
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Day 5, Summer 13
Math 5270 Transformational Geometry
14/19
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Produced
with a Trial Version of PDF Annotator - www.PDFAnno
Group
of reflections?
Day 5, Summer 13
Math 5270 Transformational Geometry
15/19
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Every isometry is a composition of reflections!
Day 5, Summer 13
Math 5270 Transformational Geometry
16/19
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Produced with a Trial Version of PDF Annotator - www.PDFAnno
Produced with a Trial Version of PDF Annotator - www.PDFAnno
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Some of students’ work
Day 5, Summer 13
Math 5270 Transformational Geometry
17/19
Congruence...
CCSS.Math.Content.HSG-CO.B.7 Use the definition of
congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are congruent.
Day 5, Summer 13
Math 5270 Transformational Geometry
18/19
Congruence...
CCSS.Math.Content.HSG-CO.B.7 Use the definition of
congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are congruent.
CCSS.Math.Content.HSG-CO.B.8 Explain how the criteria for
triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
Day 5, Summer 13
Math 5270 Transformational Geometry
18/19
Day 5, Summer 13
Math 5270 Transformational Geometry
19/19