Chapter 12 Geometric Shapes

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Chapter 12 Geometric Shapes
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Section 12.1 Recognizing Shapes
The van Hiele Theory Level 0 (Recognition) At
this lowest level, a child recognizes certain
shapes holistically without paying attention to
their components Level 1 (Analysis) The child
focuses analytically on the parts of a figure,
such as its sides and angles. Component parts and
their attributes are used to describe and
characterize figures. Relevant attributes are
understood and are differentiated from irrelevant
attributes. Level 2 (Relationships) There are
two types of thinking at this level. First, a
child understands abstract relationships among
figures. For example, a square is both a rhombus
and a rectangle. Second, a child can use informal
deductions to justify observations made at level
1. For instance, a rhombus is also a
parallelogram.
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Section 12.1 Recognizing Shapes
The van Hiele Theory Level 3
(Deduction) Reasoning at this level includes the
study of geometry as a formal mathematical
system. A student at this level can understand
the notions of mathematical postulates and
theorems. Level 4 (Axiomatics) Geometry at this
level is highly abstract and does not necessarily
involve concrete or pictorial models. The
postulates or axioms themselves become the object
of intense, rigorous scrutiny. This level of
study is only suitable for university students.
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Describing Common Geometric Shapes
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Describing Common Geometric Shapes
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Describing Common Geometric Shapes
Picture on next slide
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Steel bridge in Portland
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Describing Common Geometric Shapes
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Describing Common Geometric Shapes
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Summary
Quadrilateral
Kite
Trapezoid
Parallelogram
Isosceles Trapezoid
Rectangle
Rhombus
Square
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Section 12.2 Analyzing Shapes
Category 2
Category 1
What is the mathematical property that separates
these two categories of shapes?
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Symmetries
In formal terms, we say that an object is
symmetric with respect to a given mathematical
operation, if, when applied to the object, this
operation does not change the object or its
appearance.
Reflection Symmetry (also called folding
symmetry) A 2D figure has reflection
symmetry if there is a line that the figure can
be folded over so that one-half of the figure
matches the other half perfectly.
The fold line just described is call the
figures line (axis) of symmetry.
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Lines of symmetry for the following common
figures.
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Rotation Symmetry A 2D figure has rotation
symmetry if there is a point around which the
figure can be rotated, less than a full turn, so
that the image matches the original figure
perfectly.
(click to see animation)
This equilateral triangle has 2 (non-trivial)
rotation symmetries, 120 and 240 respectively.
Since every figure will match itself after
rotating 360, we do not consider a 360 rotation
as a rotation symmetry.
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Rotation symmetries of common figures
We dont count the trivial 360 rotation symmetry
here.
Rectangle (1 symmetry)
Square (3 symmetries)
Diamond (1 symmetry)
Parallelogram (1 symmetry)
Trapezoid (no symmetry)
Regular Pentagon (4 symmetries)
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Polygons The word "polygon" derives from
the Greek poly, meaning "many", and gonia,
meaning "angle".
Regular pentagon n 5
Equilateral triangle n 3
Square n 4
Regular hexagon n 6
Regular heptagon n 7
Regular octagon n 8
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Polygons and their nomenclature
A Triangle (from Latin) has 3 sides
A Quadrilateral (from Latin) has 4 sides
A Pentagon (from Greek) has 5 sides
A Hexagon (from Latin) has 6 sides
A Heptagon (from Greek) (or a Septagon from
Latin?) has 7 sides
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In fact, Septagon is not an official word for
the 7-gon, it is not even in a dictionary. It was
invented by some elementary school teachers to
make it easier to remember. The Latin word septem
means 7 and September means the seventh
month. The old Roman calendar began the year in
January, (named after the Roman god of fortune,
Janus), and September was the seventh month.
Afterwards, Julius Augustus (46 BC) named two
more then-29 day periods after himself and
September came to be as we know it in the
Gregorian Calendar, the ninth month.
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An Octagon (from Greek) has 8 sides
A Nonagon (from Latin) has 9 sides.
A Decagon (from Greek) has 10 sides.
A polygon with more than n (gt10) sides is usually
just called an n-gon.
More names of polygons
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Convex and Concave Shapes
A figure is convex if a line segment joining any
two points inside the figure lies completely
inside the figure.
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Angles in a polygon
In a regular pentagon the measure of a central
angle is 360/5 72 the measure of an exterior
angle is also 360/5 72 the measure of a
vertex angle is 180 72 108
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Circles
A Circle is the set of all points in the plane
that are at a fixed distance from a given point
called the center. The distance from any point
on the circle to the center is called the radius
of the circle. The length of any line segment
whose endpoints are on the circle and which
contains the center is called the diameter of the
circle. The segment is also called a diameter of
the circle.
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Circles
Circles have the following 3 properties that make
them very useful. 1. They are highly
symmetrical, hence they have a sense of beauty
and are often used in designs. eg. dinnerware.
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Circles
Circles have the following 3 properties that make
them very useful. 1. They are high symmetrical,
hence they have a sense of beauty and are often
used in designs. eg. dinnerware.
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2. Every point on a circle bears the same
distance from the center. This is called the
equidistance property. Applications wheels
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3. For a given (fixed) perimeter, the circle has
the largest area. Applications soda cans, or
any container for pressurized liquid are all
cylindrical in shape.
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Section 12.3 Properties of Lines and Angles
Definition Two different given lines L1 and L2 on
a plane are said to be parallel if they will
never intersect each other no matter how far they
are extended.
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Angles An angle is the union of two rays with a
common endpoint.
side
interior
vertex
side
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Degrees
Angles are measured by a semi-circular device
called a protractor. The whole circle is
divided into 360 equal parts, each part is
defined to have measure one degree (written 1).
Hence a semi-circular protractor has 180 degrees.
One degree is divided into 60 minutes and one
minute is further divided into 60 seconds.
Notations For instance, 27 degrees 35 minutes
41 seconds is written as 273541
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Names of angles A straight angle has 180
degrees An obtuse angle has measure between 90
and 180. A right angle has exactly 90. An
acute angle has measure less than 90.
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Definition Two angles are called vertical angles
if they are opposite to each other and are formed
by a pair of intersecting lines.
A
B
Theorem Any pair of vertical angles are always
congruent.
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More special angles
Two angles are said to be supplementary if their
measures add up to 180.
a
ß
Two angles are said to be complementary if their
measures add up to 90.
a
ß
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Perpendicular Lines Two lines are said to be
perpendicular to each other if they intersect to
form a right angle
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Parallel Lines and Angles
Definition Given two line L1 and L2 (not
necessarily parallel) on the plane, a third line
T is called a transversal of L1 and L2 if it
intersects these two lines.
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• Definitions
• Let L1 and L2 be two lines (not necessarily
  parallel) on the plane, and T be a transversal.
• ?a and ?? form a pair of corresponding angles.
• ?c and ?? form a pair of corresponding angles
  etc.

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• Definitions
• Let L1 and L2 be two lines (not necessarily
  parallel) on the plane, and T be a transversal.
• ?c and ?? form a pair of alternate interior
  angles.
• ?d and ?? form a pair of alternate interior
  angles.

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• Definitions
• Let L1 and L2 be two lines (not necessarily
  parallel) on the plane, and T be a transversal.
• ?a and ?? form a pair of alternate exterior
  angles.
• ?b and ?? form a pair of alternate exterior
  angles.

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Angle Sum in a Triangle
Draw an arbitrary triangle on a piece of paper
and label all 3 angles. Next cut out the
triangle, and then cut it into 3 parts (as
indicated by the dashed lines) Arrange the 3
angles side by side, can you get a straight angle?
b
a
c
Conclusion The angle sum in a triangle is always
180
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Angle Sum in other Polygons
What is the sum of all angles in a quadrilateral?
Answer 180? 2 360?
What is the sum of all angles in a pentagon?
Answer 180? 3 540?
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Angle Sum in other Polygons
Conclusion For a polygon with n sides, the angle
sum is (n 2) 180
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Classification of triangles according to their
angles.
A triangle with one right angle is called a right
triangle. A triangle with one obtuse angle is
called an obtuse triangle. A triangle with 3
acute angles is called an acute triangle.
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Classification of triangles according to their
sides.
A triangle with 3 different sides is called a
scalene triangle.
A triangle with 3 equal sides is called an
equilateral triangle.
A triangle with 2 equal sides is called an
isosceles triangle.
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Angles Angle Sums in Regular polygons
For a regular pentagon, m(central angle)
central angle
m(vertex angle) (3 180?) 5
108?
center
vertex angle
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Application of Degree Measure
Angles can be used to indicate directions. The
only difference is that the measure can be
greater than 180º. In navigation, the direction
can be any value between 0º and 360º.
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The Bearing System
The exact (magnetic) North is defined to be 0
degree. Any other direction is defined to be the
number of degrees away from exact North measuring
in the clockwise direction.
N
130º
south east direction
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The Bearing System
In particular, 90º is equal to exact East,
N
90º East
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The Bearing System
and 180º is equal to exact South,
N
180º South
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The Bearing System
and 270º is equal to exact West,
N
270º West
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Runway Numbers
In any airport, each runway is assigned a number
according to the direction it is pointing at
except that the units digit is omitted for
simplicity. For example, runway 24 is actually
pointing at 240º, and it means that during final
approach, the aircraft is heading 240º - which is
about south west.
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This is one of the many signs that you will see
in a big commercial airport. It tells the pilots
which runway is in front of them.
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Exercise Fill in the missing runway numbers.
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Exercise Fill in the missing runway numbers.
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Exercise Fill in the missing runway numbers.
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Exercise Fill in the missing runway numbers.
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Exercise Fill in the missing runway numbers.
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Exercise Fill in the missing runway numbers.
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Airports around San Diego

• San Diego International
• Montgomery Field
• Gillespe Field

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San Diego International
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Montgomery Field
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Section 12.4 Regular Polygons and Tessellations
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Tessellations (or Tilings) A tessellation is an
arrangement of congruent shapes that cover an
entire area with no overlaps or gaps.
A 2D geometric figure R is said to tessellate (or
tile) the plane if the entire plane can be
completely covered by (an infinite number of)
congruent copies of R with no overlaps or gaps.
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We can also tile a plane with congruent copies of
several different polygons. These are called
semiregular tessellations
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Convex and Concave Polygons
a convex quadrilateral
a concave (i.e. non-convex) quadrilateral
A polygon X is said to be convex if you take any
two points on X (including the boundary), the
line segment joining them lies entirely within
the tile (again including the boundary).
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• Question What polygons can tessellate the plane?
• Any triangle can tessellate the plane.
• Any square can tessellate the plane.
• Any rectangle can tessellate the plane.
• Any convex quadrilateral can tessellate the
  plane.
• In fact, any quadrilateral (including non-convex
  ones) can tessellate the plane.
• A regular pentagon will not tessellate the plane.
• Any regular hexagon can tessellate the plane.
• In fact, exactly 3 classes of convex hexagons can
  tile the plane.(this was proved by K. Reinhardt
  in his 1918 doctoral thesis. He also went on to
  explore the tessellations by irregular but convex
  pentagons and found 5 classes that do tile the
  plane.He felt that he had found all of them even
  though he could not give a proof because he
  claimed that it would be very tedious to do so.)

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In 1968, after 35 years working on the problem on
and off, R. B. Kershner, a physicist at Johns
Hopkins University, discovered 3 more classes of
pentagons that will tessellate. Kershner was sure
that he had found all of them, but again did not
offer a complete proof, which would require a
rather big book. Shortly after an article of
the complete classification of convex pentagons
into 8 types appeared in Scientific American
(July 1975), an amateur mathematician (R. James
III) discovered a 9th type! Between 1976 and
1977, a San Diego housewife Marjorie Rice,
without formal education in mathematics beyond
high school, found 4 more types! A 14th type was
found by a mathematics graduate student in 1985.
Since then, no new types have been found, and yet
no one knows if the classification is complete.
(the 14 types of pentagons that tile the plane)
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With the situation so intricate for convex
pentagons, you might think that it must be even
worse for polygons with 7 or more sides. However,
the situation is remarkably simple, as Reinhardt
proved in 1927 A convex polygon with 7 or
more sides cannot tessellate.
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Section 12.5 Describing 3-Dimensional Shapes
(This will be taught after section 13.2)