Download Chapter 9 Slides

Chapter 9:
Geometry
Chapter 9:Geometry
•
•
•
•
•
•
•
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
9.8 Chaos and Fractals
9.1
Points, Lines and Angles
•
•
•
•
•
Line AB
Half-line AB
Ray AB
Segment AB
Angle ABC
Types of Angles
•
•
•
•
•
•
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
•
•
•
•
•
Trapezoid
Parallelogram
Rectangle
Square
Rhombus
Circle
•
•
•
•
•
•
•
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
#76 Depth of Pond
There grows in the middle of a circular pond
10 feet in diameter a reed that projects one
foot out of the water. When it is drawn
down, it just reaches the end of the pond.
How deep is the water?
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
Euclid’s Postulates
1. Two points determine one and only one
straight line segment
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
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
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.
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.
• Angles in triangle sum
to more than 180
9.7
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
• Angles in triangle sum
to less than 180
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
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?