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10.7 Locus

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Title: 10.7 Locus


1
10.7 Locus
  • Geometry
  • Mrs. Spitz
  • Spring 2005

2
Objectives/Assignment
  • Draw the locus of points that satisfy a given
    condition.
  • Draw the locus of points that satisfy two or more
    conditions.
  • Assignment Worksheet A B
  • AND Chapter 10 Review 1-32 all.

3
Drawing a Locus that Satisfies One Condition
  • A locus in a plane is a set of all points in a
    plane that satisfy a given condition or set of
    given condition. Locus is derived from the Latin
    Word for location. The plural of locus is
    loci, pronounced low-sigh.

4
Drawing a Locus that Satisfies One Condition
  • A locus is often described as the path of an
    object moving in a plane. For instance, the
    reason many clock surfaces are circular is that
    the locus of the end of a clocks minute hand is
    a circle.

5
Ex. 1 Finding a Locus
  • Draw a point C on a piece of paper. Draw and
    describe the locus of all points that are 3
    inches from C.

6
Ex. 1 Finding a Locus
  • Draw point C. Locate several points 3 inches
    from C.

C
7
Ex. 1 Finding a Locus
  • Recognize a pattern. The points lie on a circle.

C
8
Ex. 1 Finding a Locus
  • Using a compass, draw the circle.
  • The locus of oints on the paper that are 3 inches
    from C is a circle with center C and radius of 3
    inches.

C
9
Finding a Locus
  • To find the locus of points that satisfy a given
    condition, use the following steps
  • Draw any figures that are given in the statement
    of the problem.
  • Locate several points that satisfy the given
    condition.
  • Continue drawing points until you recognize the
    pattern.
  • Draw the locus and describe it in words.

10
Loci Satisfying Two or More Conditions
  • To find the locus of points that satisfy two or
    more conditions, first find the locus of points
    that satisfy each condition alone. Then find the
    intersection of these loci.

11
Ex. 2 Drawing a Locus Satisfying Two Conditions
  • Points A and B lie in a plane. What is the locus
    of points in the plane that are equidistant from
    points A and B and are a distance of AB from B?

12
Solution
  • The locus of all points that are equidistant from
    A and B is the perpendicular bisector of AB.

13
SOLUTION continued
  • The locus of all points that are a distance of AB
    from B is the circle with center B and radius AB.

14
SOLUTION continued
  • These loci intersect at D and E. So D and E are
    the locus of points that satisfy both conditions.

15
Ex. 3 Drawing a Locus Satisfying Two Conditions
  • Point P is in the interior of ?ABC. What is the
    locus of points in the interior of ?ABC that are
    equidistant from both sides of ?ABC and 2 inches
    from P? How does the location of P within ?ABC
    affect the locus?

16
SOLUTION
  • The locus of points equidistant from both sides
    of ?ABC is the angle bisector. The locus of
    points 2 inches from P is a circle. The
    intersection of the angle bisector and the circle
    depends upon the location of P. The locus can be
    2 points OR

17
SOLUTION
  • OR 1 POINT OR NO POINTS

18
Earthquakes
  • The epicenter of an earthquake is the point on
    the Earths surface that is directly above the
    earthquakes origin. A seismograph can measure
    the distance to the epicenter, but not the
    direction of the epicenter. To locate the
    epicenter, readings from three seismographs in
    different locations are needed.

19
Earthquakes continued
  • The reading from seismograph A tells you that the
    epicenter is somewhere on a circle centered at A.

20
Earthquakes continued
  • The reading from B tells you that the epicenter
    is one of the two points of intersection of A
    and B.

21
Earthquakes continued
  • The reading from C tell you which of the two
    points of intersection is the epicenter.

epicenter
22
Ex. 4 Finding a Locus Satisfying Three
Conditions
  • Locating an epicenter. You are given readings
    from three seismographs.
  • At A(-5, 5), the epicenter is 4 miles away.
  • At B(-4, -3.5) the epicenter is 5 miles away.
  • At C(1, 1.5), the epicenter is 7 miles away.
  • Where is the epicenter?

23
Solution
  • Each seismograph gives you a locus that is a
    circle.
  • Circle A has center (-5, 5) and radius 4.
  • Circle B has center (-4, -3.5) and radius 5.
  • Circle C has center (1, 1.5) and radius 7.
  • Draw the three circles in a coordinate plane.
    The point of intersection of the three circles is
    the epicenter.

24
Solution Draw the first circle.
25
Solution Draw the second circle.
26
Solution Draw the third circle.
  • The epicenter is about (-6, 1).

27
Reminders
  • Test Tuesday/Wednesday
  • Sub on Thursday/Friday I expect all of you to
    be nice or else. Definitions/Postulates for
    Chapter 11 are due beginning of next week.
  • You will be completing 11.1 with the sub.
  • Deficiencies are due by this Friday, but all work
    you wanted considered must already be in. I will
    enter these by Wednesday.
  • Algebra Review for Chapters 9 and 10 are extra
    credit, but you must show all work.
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