Chapter 10 Introducing Geometry Section 10.2 Solving - PowerPoint PPT Presentation

1 / 23
About This Presentation
Title:

Chapter 10 Introducing Geometry Section 10.2 Solving

Description:

Chapter 10 Introducing Geometry Section 10.2 Solving Problems in Geometry Review of Basic Geometric Terms In small groups, complete the Mini-Investigation 10.2 on ... – PowerPoint PPT presentation

Number of Views:150
Avg rating:3.0/5.0
Slides: 24
Provided by: facultyShe
Category:

less

Transcript and Presenter's Notes

Title: Chapter 10 Introducing Geometry Section 10.2 Solving


1
Chapter 10Introducing Geometry
  • Section 10.2
  • Solving Problems in Geometry

2
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.

3
(No Transcript)
4
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.

5
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

6
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.

7
Example
  • Do Exercise 14 on page 545.

8
Example
  • Do Exercise 48 on page 549.

9
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.

10
(No Transcript)
11
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

12
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.

13
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.

14
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.

15
Concurrency Relationships in Triangles
  • The three altitudes are concurrent called the
    orthocenter.

16
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.

17
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.

18
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.

19
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.

20
The Euler Line
  • In every triangle, the centroid, the orthocenter,
    and the circimcenter are collinear.

21
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?

22
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.

23
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/
Write a Comment
User Comments (0)
About PowerShow.com