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Chapter 9:
Geometry
Chapter 9:Geometry
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9.1: Points, Lines, Planes and Angles
9.2: Curves, Polygons and Circles
9.3: Triangles (Pythagoras’ Theorem)
9.4: Perimeter, Area and Circumference
9.6: Transformational Geometry
9.7: Non-Euclidean Geometry, Topology
and Networks
• 9.8 Chaos and Fractals
9.1
Points, Lines and Angles
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Line AB
Half-line AB
Ray AB
Segment AB
Angle ABC
Types of Angles
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Acute
Right
Obtuse
Straight
Complementary
Supplementary
9.1
9.1
Vertical Angles
• Vertical angles have equal measure
9.1
More Angles
• Which angles are
equal?
Curves
• Simple
• Closed
9.2
Convex and Concave
9.2
Polygons
• A polygon is a simple, closed curve made
up of straight lines.
• A regular polygon is convex with all sides
equal and all angles equal.
9.2
Triangles
• Angles:
acute, right
or obtuse
• Sides:
equilateral,
isosceles,
scalene
9.2
Angle Sum of Triangle
9.2
9.2
Quadrilaterals
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Trapezoid
Parallelogram
Rectangle
Square
Rhombus
Circle
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Center
Radius
Chord
Diameter
Semicircle
Tangent
Secant
9.2
Pythagoras’ Theorem
• For a right triangle,
2
a
+
2
b
=
2
c
9.3
Proof of Pythagoras
• Add up the area of
the big square two
ways: one big square
or 4 triangles plus one
smaller square
9.3
Perimeter
• The perimeter of a
plane figure
composed of line
segments is the sum
of the measures of
the line segments, so
the total length
around the object. It
is measured in linear
units.
9.4
Area
9.4
• The area of a plane figure is the measure
of the surface covered by the figure.
Perimeter of a Triangle
• Triangle with sides
of length a, b, and
c has
P=a+b+c
9.4
Area of Triangle
• Triangle with
base b and
height h
A = ½ bh
9.4
Perimeter and Area of
Rectangle
• Rectangle with
length l and width
w has
P = 2l + 2w = 2(l + w)
A = lw
9.3
9.3
Perimeter and Area of Square
• If all sides have
length s, then
P = 4s
A = s2
Area of Parallelogram
• Parallelogram
with height h and
base b
A = bh
9.3
9.4
Area of Trapezoid
• Trapezoid with
parallel bases b
and B and height h
A = ½ h (b + B)
9.4
Circumference and Area of Circle
• Circle of radius r
has circumference
C = πd = 2πr
And area
A = πr2
9.6
Transformational Geometry
• The investigation
of how one
geometric figure
can be transformed
into another
• Reflections,
rotations,
translations and
glide reflections
Reflections
• The reflection of the point
C across the line AB is the
point C’ on the other side
of the line segment such
that CC’ is perpendicular to
AB and C and C’ are the
same distance from AB
• AB is called the line of
reflection, C’ is called the
reflection image of C
9.6
More on Reflections
9.6
• If a point A is equal to its image point A’,
then A is called an invariant point of the
transformation. The only invariant points
are on the line of reflection.
• Three or more points that lie on the same
line are said to be collinear. Reflections
preserve collinearity.
• Reflections also preserve distance.
• Use symbol rm to denote reflection across
line m
Translations
• A translation is the
composition of two
reflections across
parallel lines
• The distance between
a point and its image
is called the
magnitude of the
translation
9.6
More on Translations
• Can also describe a
translation by a line
segment AB. Each point
will move a distance equal
to the distance between A
and B, along a line
parallel to AB.
• Translations preserve
distance and collinearity.
9.6
Rotations
• A rotation is equivalent
to the composition of
two reflections about
nonparallel lines
• The point of intersection
is called the center of
rotation, angle AOA’
(where O is the center)
is the magnitude of
rotation.
9.6
More on Rotations
• Reflections preserve
collinearity and
distance
• Can also define
rotation just by the
center and magnitude
of rotation (without
the reflections)
9.6
Glide reflections
• Composition of a
translation and
reflection, where the
direction of the
translation is parallel
to the line of reflection
9.6
9.6
Isometries
• An isometry is a
transformation in
which the image
has the same
shape and size as
the original figure.
• Any isometry is
either a reflection or
the composition of
reflections.
Lines of Symmetry
• A line of symmetry of a figure is a line of
reflection for which the image of the figure
is the same figure
Euclid’s Postulates
1. Two points determine one and only one
straight line
2. A straight line extends indefinitely far in
either direction
3. A circle may be drawn with any given
center and any given radius
4. All right angles are equal
5. Given a line k and a point P not on the
line, there exists one and only one line m
through P that is parallel to k
9.7
Euclid’s Fifth Postulate
(parallel postulate)
• If two lines are such that
a third line intersects
them so that the sum of
the two interior angles is
less than two right
angles, then the two lines
will eventually intersect
9.7
Saccheri’s Quadrilateral
He assumed angles A and
B to be right angles and
sides AD and BC to be
equal. His plan was to
show that the angles C and
D couldn’t both be obtuse or
both be acute and hence
are right angles.
Non-Euclidean Geometry
9.7
• The first four postulates are much simpler than
the fifth, and for many years it was thought that
the fifth could be derived from the first four
• It was finally proven that the fifth postulate is an
axiom and is consistent with the first four, but
NOT necessary (took more than 2000 years!)
• Saccheri (1667-1733) made the most dedicated
attempt with his quadrilateral
• Any geometry in which the fifth postulate is
changed is a non-Euclidean geometry
Lobachevskian (Hyperbolic) 9.7
Geometry
• 5th: Through a point P
off the line k, at least
two different lines can
be drawn parallel to k
• Lines have infinite
length
• Angles in Saccheri’s
quadrilateral are acute
Riemannian (Spherical)
Geometry
• 5th: Through a point P off a line k, no line can
be drawn that is parallel to k.
• Lines have finite length.
• Angles in Saccheri’s quadrilateral are obtuse.
9.7
Topology
9.7
• Suppose we could study objects that could
be stretched, bent, or otherwise distorted
without tearing or scattering. This is
topology.
• Topology investigates basic structure like
number of holes or how many
components.
Topologically equivalent
• A donut and a coffee
cup are equivalent
while a muffin and
coffee cup are not.
9.7
9.7
Exercise: Letters of Alphabet
ABCDEFGH
IJKLMNOP
QRSTUVWX
YZ
9.7
Interesting Topological Surfaces
Moebius Strip
Klein Bottle
9.7
Orientability and Genus
• A topological surface is orientable if you
can determine the outside and inside.
• Any orientable, compact (finite size)
surface is determined by its number of
holes (called the genus).
9.8
Fractals
• What do we mean by
dimension? Consider
what happens when you
divide a line segment in
two on a figure. How
many smaller versions
do you get?
• Consider a line
segment, a square and
a cube.
9.8
Self-similarity
• An object is self-similar
if it can be formed from
smaller versions of itself
(with no gaps or
overlap)
• A square is self-similar,
a circle is not.
• Many objects in nature
have self-similarity.
9.8
More self-similarity in Nature
Self-similar fractals
9.8
• Start with some basic geometrical object like a
line segment or triangle and perform some
operation. Then repeat the process
indefinitely (this is called iterating). Each
iteration produces a more complicated object.
• The fractal dimension D can be found by
considering the scaling at each iteration,
where r is the scaling amount and N is the
number of smaller pieces.
rD = N so D = ln N/ln r
9.8
Cantor Set
• Start with the line
segment of length 1
between 0 and 1.
Remove the middle
third segment. Repeat
this process to the
remaining two line
segments.
• At each iteration you scale down by 3 to get 2
new pieces. What is the fractal dimension?
9.8
More on the Cantor Set
• Repeat removing middle third segments
indefinitely. How much length is left?
Sierpinski Gasket
9.8
• Start with an equilateral triangle. Divide
each side in half and remove the middle
triangle. Repeat this process indefinitely.
Sierpinski Gasket
9.8
• What happens to the
perimeter as you do
more iterations?
• What about area?
• What is the fractal
dimension of the
gasket? Does this
make sense?
9.8
Koch Snowflake
• Start with equilateral
triangle. Iteration rule:
• What happens to the
perimeter? Area?
• What is the fractal
dimension?