10'2 Arcs and Chords

1
10.2 Arcs and Chords

• Standard 21.0 Students prove and solve problems
  regarding relationships among chords, secants,
  tangents, inscribed and circumscribed polygons of
  circles

2
2-3 minute activity with compass and protractor.

• Draw a circle. Draw an angle whose vertex is the
  center of the circle.
• Label it central angle.
• Measure your central angle.
• Who has an angle less than 180o?
• Color or highlight the part of the circumference
  that the angle intersects the circle at two
  points of the circle as the minor arc.
• Who has an angle more than 180o?
• Color or highlight the part of the circumference
  that the angle intersects the circle at two
  points of the circle as the major arc.

3
Measuring Arcs

• The measure of a minor arc, is defined to be the
  measure of its central angle.
• mGFmltGHF60o
• The measure of the major arc, is defined as the
  difference of
• 360o associated
• minor arc.
• mGFmltGEF360-60300o

G
60o
60o
E
F
H
4
Goal 1 Using Arcs of circles.

• Central angle-In a plane, an angle whose vertex
  is the center of a circle.
• Minor Arc, if the measure of the central angle is
  less than 180o.
• Major arc- if the measure of the central angle is
  less than 180o
• If the endpoints of an arc are the endpoints of
  the diameter, then the arc is a semicircle.

5
Naming arcs

• The minor arc associated with
• ltAPB is AB.
• The major arcs and
• Semicircles are named by their
• Endpoints and by a pt. on the
• Arc. The major arc is
• ACB.

Minor arc ltAPB
A
B
P
C
Major arc ltAPB in the exterior of the central
angle.
6
2-3 minute activity with protractor and compass

• Draw a circle H. Draw a diameter and lable it as
  EF. Find another point on the circle, label it
  G. Then draw a radius to this point. Find the
  mGF and the mGEF, mEG, and mEF .
• EGF is a semicircle, label it.
• Is there a relationship between mEG, mFG, and
  mEGF?

7
Arc Addition postulate

• The measure of an arc formed by two adjacent arcs
  is the sum of the measures of the two arcs.
• mEGFmEGmGF

G
60o
60o
E
F
H
8
Example 1
9
Example 2
10
2-3 minute activity with a partner

• Draw a circle, then draw two chords of the same
  length that start at the same point on the circle
  and end at other points on the circle. Label the
  starting point B. Label the other points of the
  chord that meet the circle point A and C
  respectively. Use tick marks to denote that the
  chords are congruent. Now draw a radius from the
  center to point A and C. Measure the arcs
• AB and BC. What is their relationship?
• Conjecture? In the same circle, or in congruent
  circles two minor arcs are congruent if?

11
Chord Theorems 10.4

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

A
C
B
12
2-3 minutes and I will call on something? When I
call on someone your partner will answer.

• Use a protractor and a compass. Draw a circle
  with a compass, draw a diameter. How do you know
  that it is a diameter? Draw a perpendicular
  chord that intersects the diameter. What did you
  make?
• Chord?
• What is the relationship between the 2 pieces of
  the chord and the diameter(use a ruler)? What is
  the relationship between the two pieces of the
  arcs that is created from the chord and split by
  the diameter.(hint use a protractor)
• Make a conjecture?If a diameter of a circle is
  perpendicular to the chord then?

13
Chord Theorem 10.5 and 10.6
14
2-3 minute activity

• Draw a circle E. Draw two nonintersecting chords
  of the same length. Label the segment of one
  chord AB and the other CD. Draw a line from the
  center to a point on the chord that is
  perpendicular to chord AB. Draw a line from the
  center to a point on the chord that is
  perpendicular to chord CD. Measure the lengths of
  each of lines that start from point E to the
  chord.
• Conjecture- If the same circle or congruent
  circles, two chords are congruent if and only if

15
Theorem 10.7
16
Day 2
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Proof of 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.
• Page 610 question number 59.

18
Proof of theorem 10.6

• Page 610 question number 61
• Theorem 10.6 states If one chord is a
  perpendicular bisector of another chord,
• then the first chord is a diameter.