Download Geometric Isometries

Math 416
Geometry Isometries
Topics Covered
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1) Congruent Orientation – Parallel Path
2) Isometry
3) Congruent Relation
4) Geometric Characteristic of Isometry
5) Composite
6)Geometry Properties
7) Pythagoras – 30 - 60
Congruent Figures
• Any two figures that are equal in every
aspect are said to be congruent
• Equal is every aspect means…
–
–
–
–
All corresponding angles
All corresponding side lengths
Areas
Perimeters
Congruent Figures
• We also note that we are talking
about any figures in the plane not
just triangles
• However, it seems in most geometry
settings, we deal with triangles
• We hope this section will allow you to
look at all shapes but…
Orientation
• One of the most important
characteristics of shapes in the plane
is its orientation
• How the shape is oriented means the
order that corresponding point
appear
Orientation
• Consider
A
A’
#1
B
#2
C
B’
C’
Orientation
• To establish the order of the points, we
need two things;
• #1) A starting point – that is a
corresponding point
• #2) A direction – to establish order
• “Consistency is the core of mathematics”
• I will choose A and A’ as my starting points
Orientation
• I will choose
counterclockwise
as my direction
• Hence in triangle
#1 we have A -> B >
C.
• In Triangle #2 we
have A’ -> B’ -> C’
Orientation
• Since the corresponding points
match, we say the two figures have
the same orientation.
• Consider A
A’
#1
B
#2
C
C’
B’
Orientation Vocabulary
• These figures do not have the same
orientation
• Same orientation can be phrased as
follows;
– orientation is preserved
- orientation is unchanged
- orientation is constant
Orientation Vocabulary
• Different Orientation can be stated
- orientation is not preserved
- orientation is changed
- orientation is not constant
Parallel Path
• We are interested how one congruent
figure gets to the other
• We are interested how one congruent
figure is transformed into another
• We call the line joining corresponding points its path
• i.e. A A’ is the path
• If we look at all the paths between corresponding
points, we can determine if all the paths are parallel.
Examples
A’
A
B’
C’
B
C
These are a parallel path
Examples
A’
A
C’
B’
B
C
These are not parallel paths
It is called Intersecting Paths
Types of Isometries
• There are 4 Isometries
1) Translation
2) Rotation
3) Reflection
4) Glide Reflection
Translation
• Translation – moving points of a
figure represented by the letter t.
• As you may recall t (-2,4) (x – 2, y + 4)
You move on the x axis minus 2 and
on the y axis you move plus 4.
Rotations
• Rotations: Rotations can be either
90, 180, 270 or 360 degrees.
• Rotations can be clockwise or
counter-clockwise
• Represented by the letter r
Reflection
• You can have reflections of x
• You can have a reflections of y
Glide Reflection
• Glide reflection occurs when the
orientation is not preserved AND
does not have a parallel path.
• Can be best seen with examples…
Tree Diagram
• We can define the four isometries by
the way of these two characteristics
Orientation Same? Parallel Path?
YES
YES
No
YES
No
No
TRANSLATION
ROTATION
REFLECTION
GLIDE REFLECTION
Table Representation
With Parallel
Path
Without
Parallel Path
Orientation
Same
(maintained)
Translation
Orientation
Different
(changed)
Reflection
Rotation
Glide
Reflection
Notes
• The biggest problem is establishing corresponding
points.
• It is easy when they tell you AA’, BB’ but it is
usually not the case 
• Let’s try two examples… what kind of isometric
figures are these…
•
You may choose to cut up the figure on a piece
of paper which can help locate the points…
•
Example
#1
Consider (we assume they are congruent)
Bigger
Angle
90°
Bigger
Angle
Smaller
Angle
90°
Smaller
Angle
• We need to establish the points. Look for
clues (bigger, 90 and smaller angle).
Which Isometric Figure?
ORIENTATION? PARALLEL PATH?
• Hence orientation ABC A’C’B’ are
NOT the same…
• Parallel paths… No!
A
B
A’
B’
C
C’
GLIDE REFLECTION
Example #2
A
B
C’
C
B’
A’
ORIENTATION? PARALLEL PATHS?
ABC and A’B’C’ – Orientation the same
Not Parallel Paths
ROTATION
Other Figures
• When the figure is NOT a triangle,
you can usually get away with just
checking three points. The hard part
is finding them. Let’s take a look at
two more examples
Example with a Square
° A
B
C
Glide
Reflection
°
A’
B’
C’
Orientation / Parallel Paths?
Orientation Changed, Not Parallel
Practice
°
°
Orientation? Parallel?
90o
counter
clockwise
rotation
Orientation Same; Not
Parallel  Rotation
The Congruency Relation
• When we know two
shapes are
congruent (equal),
we use the symbol.
Congruent
Symbol
Congruency Relation
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•
•
•
•
•
Hence if we say HGIJ
We note
H corresponds to K
G corresponds to L
I corresponds to M
J corresponds to N
KLMN
Congruency Relation
• From this we state the following
equalities.
• Line length
• HG = KL (1st two)
• GI = LM (second two)
• IJ = MN (last two)
• HJ = KN (outside two)
Congruency Relation
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•
Angles
< HGI = < KLM (1st two)
< GIJ = < LMN (second two)
< IJH = < MNK (last two first)
< JHG = < NKL (last one 1st two)
We have established all this without
seeing the figure!
Exam Question
• State the single isometry. State
the congruency relation and the
resulting equalities.
A
C
B
D
K
L
M
N
Hence BACD KMNL
Exam Question
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We can also can note that…
BK
DL
CN
A  M Clockwise
Orientation / Parallel Path?
Exam Solution
• Orientation Changed
• Parallel Path
• Reflection
Other Findings
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Line Length
BA = KM
AC = MN
CD = NL
DB = LK
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Angles
< BAC = < KMN
< ACD = < MNL
< CDB = < NLK
< DBA = < LKM
Test Question
Given ABCDE
FGHIJ
A
E
B
True or False?
D
C
•You should draw a
F
J
G
diagram to clarify…
I
H
< ABC = HIJ False
< ABC = HGF True
BC = HI False
Two Isometries – Double
the fun!
• At certain points,
we may impose
more than one
isometry.
• Consider
1
2
3
We say 1  2 is a reflection of
2  3 is a rotation r
s
Notes
• We would say that the composite is
r
°
s
after
We can say there is a
rotation after a reflection.
So you should read from
right to left
Notes
• We also note that 1 – 3 is a glide
reflection (gr)
• Hence r ° s = gr
2
1
Math
is fun
3
Math
is fun
1  2 t
2  r
Thus r ° t = r
Math
is fun
• Consider
Practice
Geometry Reminders
Complimentary Angles
• Here are some reminders of things
you should know.
b
a
Complimentary
angles add up to
90o. Thus <a + <b
= 90o
Supplementary Angles
Supplementary angles add up to
180o. All straight lines form an
angles of 180o. Thus <a + < b =
180o
a
b
Vertically Opposite
Angles
a
d
b
c
Vertically
opposite angles
are equal.
Thus <a = <c
and <b = <d
Isoscelles Triangles
x
x
The angles
opposite the
equal sides are
equal or vice
versa
Angles in a Triangle
a
b
c
Angles in a
triangle add
up to 180o.
Thus <a +
<b + <c =
180o.
Parallel Lines
Transversal
Line
c
w
y
z
x
a
d
b
When a line
(transversal)
crosses two parallel
lines, four angles
are created at
each line
Parallel Lines
• The following relationship between each
group is created.
• Alternate Angles
- both inside (between lines) & the
opposite side of tranversal are EQUAL.
Thus, < c = < x
a b
<d=<w
c
d
w
y
z
x
Corresponding Angles
• Both same side of tranversal one
between parallel lines the other
outside parallel lines are EQUAL
• <a = <w
a b
• <c <y
c d e
• <b = < x
w
• <d = <z
x
y
z
<b & <e are called
alternate interior angle
Supplemental Angles
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•
•
•
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Both same side of transversal
Both between parallel lines
Add up to 180°
Therefore, <c + <w = 180°
<d + <x = 180°
Practice
G
5x+35
A
C
We note < DEB
= < ABG
(corresponding)
<DEB = <HEF
(vertical)
B
E
D
H
2x + 92
F
5x+35=2x+92
3x = 57
X = 19
Solution
G
130
A
C
Replace x = 19
into 5x+35
B
D
130 50
130
H
5(19) + 35
F
= 130
Test Question
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What is the angle < ABC?
A
5x + 3 + 2x - 20 + x + 5 = 180
8x -12 = 180
5x+3
B
8x = 192
x = 24
2x-20
Replace x = 24 into 2x – 20
2 (24) – 20
x+5
= 28°
C
Pythagoras Theorem
• The most famous and most used
theorem or geometric / algebraic
relationship is Pythagoras Theorum
• In words – the square of the
hypotenuse is equal to the sum of the
square on the of the other two sides
Pythagoras Example
• Which of these numbers (3,4,5) must
be the hypotenuse? Establish 90°
5
3
4
• Does the placement of the 3, 4 or 5
make a difference?
•
Formula c2 = a2 + b2
•
Have one unknown. Solve and switch
for practice
Pythagoras in Geometry
• If we have a right angle triangle with
a 30° (or a 60°)
• The side opposite the 30° angle is
half the hypotenuse
• Or.. the hypotenuse is twice the side
opposite the 30° angle
Practice
½x
Hence if the
hypotenuse is 8, x = ?
x
x = 4
30°
or
x
30°
Practice
5
60°
x
y
x = ?
y = ?
x = 10
102 = 52 + y2
100 = 25 + y2
75 = y2
y =8.66