Transcript Document 7194387

Section 6.1
Circles and Related Segments and Angles
• A circle is the set of all points in a plane that are at a
fixed distance from a given point known as the center of
the circle.
• A circle is named by its center point.
• The symbol for circle is 
• A radius is a segment that joins the center of the circle
to a point on the circle.
• All radii of a circle are congruent
• A line segment that joins two points of a circle is a
chord.
• A diameter of a circle is a chord that contains the center
of the circle
5/20/2016
Section 6.1 Nack
1
Segments of a Circle
• What are the radii of Q?
• QS, QW, QV, QT
• What is the diameter?
• WT
• What is the chord?
• SW, TW
• Congruent circles are two or
more circles that have
congruent radii.
5/20/2016
Section 6.1 Nack
2
Relationships Concerning Circles
• Concentric Circles are
coplanar circles that have
a common center.
O
• An arc is a segment of a circle
determined by two points on the circle
and all the points in between. ABC is
an arc denoted by ABC.
• A semicircle is the arc determined by a
diameter
– A minor arc is part of a semicircle.
– A major arc is more than a
semicircle but less than a circle
D
• Theorem 6.1.1: A radius
that is perpendicular to a
chord, bisects the chord.
A
P
B
5/20/2016
Section 6.1 Nack
C
3
Angles and Arcs
• A central angle of a circle
is an angle whose vertex is
the center of the circle and
whose sides are radii.
• An intercepted arc is
determined by the two
points of intersection of the
angle with the circle and all
points of the arc in the
interior of the angle.
ROT is a central angle of
O.
RT is the intercepted arc.
5/20/2016
Section 6.1 Nack
4
Angle and Arc Relationships in the Circle
• Postulate 16: (Central Angle Postulate): In a circle, the
degree measure of a central angle is equal to the degree
measure of the intercepted arc.
• Congruent arcs are arcs with equal measure in either a
circle or congruent circles.
mAOB = mCOD
m AEB = mCFD
C
A
E
F
O
B
5/20/2016
Section 6.1 Nack
D
5
• Postulate 17: (Arc-Addition Postulate): If B lies between A and C on
a circle, then mAB +mBC = mABC An arc is equal to the sum of its parts.
• Inscribed angle of a circle whose vertex is a point on the circle and
whose sides are chords of the circle.
RST is the inscribed angle with chords SR and ST as the sides.
• Theorem 6.1.2: The measure of an inscribed angle of a
circle is one-half the measure of its intercepted arc.
Case 1: One chord is a diameter (below)
Case 2: The diameter lies inside the inscribed angle p. 281 a
Case 3: The diameter lies outside the inscribed angle p. 281 b
Proof Case 1: Draw the radius RO. ROS is isoceles
with mR = m S.
Since ROT is the exterior angle of ROS then
mROT = mR + m S = 2 mS
Therefore, mS = ½ mROT.
Since the measure of the arc is equal to the measure
of the angle then by substitution mS = ½ m RT.
5/20/2016
Section 6.1 Nack
6
More on Chords and Arcs
• In a circle (or in congruent circles),
– Theorem 6.1.3: congruent minor arcs have congruent
central angles p. 282.
– Theorem 6.1.4: congruent central angles have
congruent arcs.
– Theorem 6.15: congruent chords have congruent
minor (major) arcs.
– Theorem 6.16: congruent arcs have congruent
chords.
C
A
O
P
B
5/20/2016
D
Section 6.1 Nack
7
• Theorem 6.1.7: Chords that are at the same distance
from the center of a circle are congruent. Figure 6.18
• Theorem 6.1.8: Congruent chords are located at the
same distance from the center of the circle. Figure 6.18
• Theorem 6.1.9: An angle inscribed in a semicircle is a
right angle. Figure 6.19 p. 282
• Theorem 6.1.10: If two inscribed angles intercept the
same arc, then these angles are congruent. Figure 6.20
5/20/2016
Section 6.1 Nack
8