10.2 Arcs and Chords

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10.2 Arcs and Chords

• Geometry
• Mrs. Spitz
• Spring 2005

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Objectives/Assignment

• Use properties of arcs of circles, as applied.
• Use properties of chords of circles.
• Assignment pp. 607-608 3-47
• Reminder Quiz after 10.3 and 10.5

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Using Arcs of Circles

• In a plane, an angle whose vertex is the center
  of a circle is a central angle of the circle. If
  the measure of a central angle, ?APB is less than
  180, then A and B and the points of P

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Using Arcs of Circles

• in the interior of ?APB form a minor arc of the
  circle. The points A and B and the points of
  P in the exterior of ?APB form a major arc of
  the circle. If the endpoints of an arc are the
  endpoints of a diameter, then the arc is a
  semicircle.

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Naming Arcs

• Arcs are named by their endpoints. For example,
  the minor arc associated with ?APB above is
  . Major arcs and semicircles are named by their
  endpoints and by a point on the arc.

60
60
180
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Naming Arcs

• For example, the major arc associated with ?APB
  is .
• here on the right is a semicircle.
• The measure of a minor arc is defined to be the
  measure of its central angle.

60
60
180
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Naming Arcs

• For instance, m m?GHF 60.
• m is read the measure of arc GF. You can
  write the measure of an arc next to the arc. The
  measure of a semicircle is always 180.

60
60
180
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Naming Arcs

• The measure of a major arc is defined as the
  difference between 360 and the measure of its
  associated minor arc. For example, m
  360 - 60 300. The measure of the whole
  circle is 360.

60
60
180
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Ex. 1 Finding Measures of Arcs

• Find the measure of each arc of R.

80
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Ex. 1 Finding Measures of Arcs

• Find the measure of each arc of R.
• Solution
• is a minor arc, so m m?MRN 80

80
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Ex. 1 Finding Measures of Arcs

• Find the measure of each arc of R.
• Solution
• is a major arc, so m 360 80 280

80
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Ex. 1 Finding Measures of Arcs

• Find the measure of each arc of R.
• Solution
• is a semicircle, so m 180

80
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Note

• Two arcs of the same circle are adjacent if they
  intersect at exactly one point. You can add the
  measures of adjacent areas.
• Postulate 26Arc Addition Postulate. The measure
  of an arc formed by two adjacent arcs is the sum
  of the measures of the two arcs.

m m m
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Ex. 2 Finding Measures of Arcs

• Find the measure of each arc.
• m m m
• 40 80 120

40
80
110
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Ex. 2 Finding Measures of Arcs

• Find the measure of each arc.
• m m m
• 120 110 230

40
80
110
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Ex. 2 Finding Measures of Arcs

• Find the measure of each arc.
• m 360 - m
• 360 - 230 130

40
80
110
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Ex. 3 Identifying Congruent Arcs

• Find the measures of the blue arcs. Are the arcs
  congruent?

45
45
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Ex. 3 Identifying Congruent Arcs

• Find the measures of the blue arcs. Are the arcs
  congruent?

80
80
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Ex. 3 Identifying Congruent Arcs

• Find the measures of the blue arcs. Are the arcs
  congruent?

65

• m m 65, but and are
  not arcs of the same circle or of congruent
  circles, so and are NOT
  congruent.

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Using Chords of Circles

• A point Y is called the midpoint of if
  ? . Any line, segment, or ray that
  contains Y bisects .

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Theorem 10.4

• In the same circle, or in congruent circles, two
  minor arcs are congruent if and only if their
  corresponding chords are congruent.

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Theorem 10.5

• If a diameter of a circle is perpendicular to a
  chord, then the diameter bisects the chord and
  its arc.

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Theorem 10.5

• If one chord is a perpendicular bisector of
  another chord, then the first chord is a diameter.

is a diameter of the circle.
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Ex. 4 Using Theorem 10.4
(x 40)

• You can use Theorem 10.4 to find m .

2x
2x x 40
Substitute
Subtract x from each side.
x 40
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Ex. 5 Finding the Center of a Circle

• Theorem 10.6 can be used to locate a circles
  center as shown in the next few slides.
• Step 1 Draw any two chords that are not
  parallel to each other.

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Ex. 5 Finding the Center of a Circle

• Step 2 Draw the perpendicular bisector of each
  chord. These are the diameters.

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Ex. 5 Finding the Center of a Circle

• Step 3 The perpendicular bisectors intersect at
  the circles center.

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Ex. 6 Using Properties of Chords

• Masonry Hammer. A masonry hammer has a hammer on
  one end and a curved pick on the other. The pick
  works best if you swing it along a circular curve
  that matches the shape of the pick. Find the
  center of the circular swing.

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Ex. 6 Using Properties of Chords

• Draw a segment AB, from the top of the masonry
  hammer to the end of the pick. Find the midpoint
  C, and draw perpendicular bisector CD. Find the
  intersection of CD with the line formed by the
  handle. So, the center of the swing lies at E.

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Theorem 10.7

• In the same circle, or in congruent circles, two
  chords are congruent if and only if they are
  equidistant from the center.
• AB ? CD if and only if EF ? EG.

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Ex. 7 Using Theorem 10.7

• AB 8 DE 8, and CD 5. Find CF.

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Ex. 7 Using Theorem 10.7

• Because AB and DE are congruent chords, they are
  equidistant from the center. So CF ? CG. To
  find CG, first find DG.
• CG ? DE, so CG bisects DE. Because DE 8, DG
  4.

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Ex. 7 Using Theorem 10.7

• Then use DG to find CG. DG 4 and CD 5, so
  ?CGD is a 3-4-5 right triangle. So CG 3.
  Finally, use CG to find CF. Because CF ? CG, CF
  CG 3

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Reminders

• Quiz after 10.3
• Last day to check Vocabulary from Chapter 10 and
  Postulates/Theorems from Chapter 10.