Transcript 10.6 Circles and Arcs - Cardinal O'Hara High School

10.6 Circles and Arcs
• In a plane, a circle is the set of all points equidistant
from a given point called the center.
• A diameter is a segment that contains the center of
a circle and has both endpoints on the circle.
• A radius is a segment that has one endpoint at the
center and the other endpoint on the circle.
• Congruent circles have congruent radii.
• A central angle is an angle whose vertex is the
center of the circle.
• Way to write the name of a circle:
P
P
Circles
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An arc is a part of a circle.
A semicircle is one type of arc – half of a circle.
A minor arc is smaller than a semicircle.
A major arc is larger than a semicircle.
You name a minor arc by its endpoints and a major arc or a
semicircle by its endpoints and another point on the arc.
Naming Arcs
• What are the minor arcs of circle O?
AD, CE, AC, DE
• What are the semicircles of circle O?
ACE, CED, EDA, DAC
• What are the major arcs of circle O
that contain point A?
ACD, CEA, EDC, DAE
Arc Measure
• The measure of a minor arc is equal to the measure
of its corresponding central angle.
• The measure of a major arc is the measure of the
related minor arc subtracted from 360.
• The measure of a semicircle is 180.
Adjacent Arcs
• Adjacent arcs are arcs of the same circle that
have exactly one point in common.
• You can add the measures of adjacent arcs just
as you can add the measures of adjacent
angles.
Arc Addition Postulate
• The measure of the arc formed by two
adjacent arcs is the sum of the measures of
the two arcs.
mABC  mAB  mBC
Finding the Measures of Arcs
• What is the measure of each arc in circle O?
A. BC
B. BD
C. ABC
D. AB
mBC  mBOC  32
mBD  mBC  mCD
mBD  32  58  90
mABC  180
mAB  180  32  148
Circumference
• The circumference of a circle is the distance
around the circle.
• The number pi ( ) is the ratio of the
circumference of a circle to its diameter.
C d
C  2 r
22
  3.14OR
7
Concentric Circles
• Coplanar circles that have the same center are
concentric circles.
Arc Length
• The measure of an arc is in degrees while the arc
length is a fraction of a circle’s circumference.
Finding Arc Length
• What is the length of the arc shown in red? Leave your
answer in terms of
.
mXY
lengthof XY 
 d
360
90

  (16)
360
 4 in.
mXPY
lengthof XPY 
 2 r
360
240

 2 (15)
360
 20 cm
Congruent Arcs
• Congruent arcs are arcs that have the same
measure and are in the same circle or in
congruent circles.
More Practice!!!!!
• Homework – p. 654 # 9 – 23.
• Homework – p. 654 – 655 # 24 – 35.