Chapter 10 Introducing Geometry

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Chapter 10Introducing Geometry

• Section 10.2
• Solving Problems in Geometry

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Review of Basic Geometric Terms

• In small groups, complete the Mini-Investigation
  10.2 on page 534 of your textbook.
• Be sure to use appropriate symbols when naming
  each.

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Point and Line Problems

• Points and lines are basic building blocks for
  other geometric ideas and relationships.
• Network (graph) theory is very useful for solving
  some types of real-world problems.

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Vocabulary of Networks

• Vertices points of intersection
• Edges segments or arcs
• Network total configuration of vertices and
  edges
• Traversable Network paths of network can be
  traced only once without lifting pencil from
  paper
• Traversable Type 1 beginning and ending point
  is same
• Traversable Type 2 beginning and ending points
  do not coincide
• Odd Vertex odd number of edges meet at vertex
• Even Vertex even number of edges meet at vertex

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Network Traversability Theorem

• If a network has all even vertices, it is
  traversable type 1, and the traversable path may
  begin at any vertex.
• If a network has exactly two odd vertices, it is
  traversable type 2, and the traversable path must
  begin at one of the odd vertices and end at the
  other.
• If a network has more than two odd vertices, it
  is not traversable.

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Example

• Do Exercise 14 on page 545.

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Example

• Do Exercise 48 on page 549.

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Review of Ideas about Segments, Angles, and
Triangles

• In small groups, complete the Mini-Investigation
  on page 537 of your textbook.
• Be sure to use appropriate symbols when naming
  each.

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Triangle Concurrency Activity

• Using the sheet of triangles provided in class,
  cut out each triangle carefully.
• We will use each triangle to fold segments that
  will represent the following segments in a
  triangle
• Median
• Altitude
• Perpendicular Bisectors
• Angle Bisectors

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Median of a Triangle

• A median of a triangle is a segment that extends
  from the vertex of an angle to the midpoint of
  the opposite side.

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Concurrency Relationships in Triangles

• The three medians are concurrent at a point
  called the centroid, which is the balance point
  or center of gravity, of the triangle. The
  centroid is two-thirds the distance from each
  vertex to the opposite side.

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Altitude of a Triangle

• An altitude of a triangle is a segment that
  extends from the vertex of an angle and is
  perpendicular to the opposite side.

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Concurrency Relationships in Triangles

• The three altitudes are concurrent called the
  orthocenter.

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Perpendicular Bisector of a Triangle

• A perpendicular bisector of a triangle is a
  segment (or line) that is perpendicular to each
  side at its midpoint.

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Concurrency Relationships in Triangles

• The three perpendicular bisectors of the sides
  are concurrent at a point called the
  circumcenter, which is the center of a circle
  containing the vertices. We say that this circle
  circumscribes the triangle and that the triangle
  is inscribed in the circle.

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Angle Bisector of a Triangle

• An angle bisector of a triangle is a segment that
  bisects each angle at its vertex and extends to
  the opposite side.

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Concurrency Relationships in Triangles

• The three angle bisectors are concurrent in a
  point called the incenter, which is the center of
  a circle that is tangent to each side of the
  triangle. We say that this circle is inscribed
  in the triangle.

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The Euler Line

• In every triangle, the centroid, the orthocenter,
  and the circimcenter are collinear.

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An Application to Earthquakes

• During a recent earthquake in the western United
  States, three cities recorded identical measures
  of intensity from seismographic readings taken at
  the same time. Since the seismographic readings
  were the same, each city was the same distance
  from the epicenter of the earthquake.
• How could the epicenter of the earthquake be
  determined using this information?

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Earthquake Extension Activity

• Since seismographic readings are seldom the same
  from city to city, seismologists use circles to
  pinpoint the location of the epicenter of an
  earthquake.
• Readings from cities are taken and an epicentral
  distance from the readings are used as a radius
  of a circle that is constructed around each city.
• The intersection of the circles pinpoints the
  location of the earthquakes epicenter.

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Earthquake Extension Activity

• Students can learn how actual seismologists
  determine the epicenter and intensity of an
  earthquake by visiting a Virtual Earthquake site
  and doing an online activity.
• Web Address http//sciencecourseware.com/Virtual
  Earthquake/