Math 416 Geometry Isometries. Topics Covered 1) Congruent Orientation – Parallel Path 2) Isometry...
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Transcript of Math 416 Geometry Isometries. Topics Covered 1) Congruent Orientation – Parallel Path 2) Isometry...
Math 416Math 416Math 416Math 416
Geometry Isometries Geometry Isometries
Topics Covered• 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
#1 #2
A A’
B 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 #2
B 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
These are a parallel path
A
C’
C
B’
B
A’
Examples
A
BC
B’
A’
C’
These are not parallel paths
It is called Intersecting Paths
Types of Isometries• There are 4 Isometries
1) Translation2) Rotation3) Reflection4) 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
YES
No
No
No
TRANSLATION
ROTATION
REFLECTION
GLIDE REFLECTION
Table RepresentationOrientation Same (maintained)
Orientation Different (changed)
With Parallel Path
Translation Reflection
Without Parallel Path
Rotation Glide Reflection
Notes• The biggest problem is establishing
corresponding points.
• It is easy when they tell you AA’, BB’ 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)
• We need to establish the points. Look for clues (bigger, 90 and smaller angle).
90°
90°Bigger Angle
Bigger Angle
Smaller Angle
Smaller Angle
Which Isometric Figure?
• Hence orientation ABC A’C’B’ are NOT the same…
• Parallel paths… No!A
C’
C
B’
B
A’
GLIDE REFLECTION
ORIENTATION? PARALLEL PATH?
Example #2 A
B’
CB
C’
A’
ABC and A’B’C’ – Orientation the same
ORIENTATION? PARALLEL PATHS?
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
°
°
B
C
B’
C’
A’
A
Orientation / Parallel Paths?
Orientation Changed, Not Parallel
Glide Reflectio
n
Practice
°°
Orientation? Parallel?
Orientation Same; Not Parallel Rotation
90o counter clockwi
se rotatio
n
The Congruency Relation
• When we know two shapes are congruent (equal), we use the symbol.
CongruentSymbol
Congruency Relation• Hence if we say HGIJ KLMN• We note • H corresponds to K• G corresponds to L• I corresponds to M• J corresponds to N
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• 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
DC
B K
L N
M
Hence BACD KMNL
Exam Question• 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• Line Length• BA = KM• AC = MN• CD = NL• DB = LK
• Angles• < BAC = < KMN• < ACD = < MNL• < CDB = < NLK• < DBA = < LKM
Test QuestionGiven ABCDE FGHIJ
True or False?
•You should draw a diagram to clarify…
False
A
D C
B
IGJ
H
F
< ABC = HIJ
E
< ABC = HGF True
BC = HI False
Two Isometries – Double the fun!
• At certain points, we may impose more than one isometry.
• Consider 1 2
We say 1 2 is a reflection of s
3
Math
#
1Math
#1
Math #1
2 3 is a rotation r
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
Practice• Consider
1 2 3
1 2 t
2 r
Thus r ° t = r
Math is fun
Math
is
fu
n
Math is fun
Geometry RemindersComplimentary Angles
• Here are some reminders of things you should know.
ba
Complimentary angles add up to 90o. Thus <a +
<b = 90o
Supplementary Angles
a b
Supplementary angles add up to 180o. All straight lines form an angles of 180o. Thus
<a + < b = 180o
Vertically Opposite Angles
a
bd
c
Vertically opposite
angles are equal. Thus <a = <c and
<b = <d
Isoscelles Triangles
The angles opposite the equal sides are equal or vice versa
x x
Angles in a Triangle
a
b c
Angles in a triangle
add up to 180o. Thus <a + <b + <c = 180o.
Parallel Lines
a bc d
xw
y z
When a line (transversal) crosses two
parallel lines, four angles are created at each
line
Transversal 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 < d = < w
a bc d
xw
y z
Corresponding Angles• Both same side of tranversal one
between parallel lines the other outside parallel lines are EQUAL
• <a = <w• <c <y• <b = < x• <d = <z
a bc d
xw
y z
e
<b & <e are called alternate interior
angle
Supplemental Angles• Both same side of transversal • Both between parallel lines• Add up to 180°• Therefore, <c + <w = 180° • <d + <x = 180°
Practice
5x+35
2x + 92
A
D F
C
G
H
B
We note < DEB = < ABG
(corresponding)
<DEB = <HEF (vertical)
E
5x+35=2x+92
3x = 57
X = 19
130
130
A
D F
C
G
H
B
50130
Solution
Replace x = 19 into 5x+355(19) +
35
= 130
Test Question• What is the angle < ABC?• 5x + 3 + 2x - 20 + x + 5 = 180• 8x -12 = 180• 8x = 192• x = 24• Replace x = 24 into 2x – 20• 2 (24) – 20• = 28°
5x+3
2x-20
x+5
A
C
B
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) mustbe 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 x
30°
Hence if the hypotenuse is 8, x
= ?x = 4
or 2x
x30°
Practice
5 x60° x = ?
x = 10y
y = ?
102 = 52 + y2
100 = 25 + y2
75 = y2
y =8.66