warm-up find the measure of each arc of circle z. a. mbd b. mace c. mdeb d. mabc 140° 230° 130° z...
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Warm-UpFind the measure of each arc of circle Z.
a. mBD
b. mACE
c. mDEB
d. mABC
140°
230°
130°Z
D
55°
BA
105°90°
75°
35°
C
E
270°
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Section 10 – 3 & Section 10 – 6
Apply Properties of Chords
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Theorem 10-3In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
A
B C
D
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Theorem 10-4If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
S P
T
Q
R
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Theorem 10-5If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
S P
T
Q
R
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Theorem 10-6In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
A
P
C
G
DB
F
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Theorem 10-14 Segments of Chords Theorem
If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
D
A
C
B
E
EA • EB = EC • ED
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Example 1In circle R, AB = CD and mAB = 108°. Find mCD.
Since AB and CD are = chords, then AB = CD.
mCD = 108°
R
B
108°
CA
D
~
~~
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Example 2Use the diagram of Circle C to find the length of BF. Tell what theorem you used.
10 units
Theorem 10-5: If a diameter is perpendicular to a chord than the diameter bisects the chord and its arc.
C
G
A
BDF
10
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Example 3In the diagram of Circle P, PV = PW, QR = 2x + 6 and ST = 3x – 1. Find QR.
Use Theorem 10-6
x = 7
QR = STR
V S
PQ
T
W 2x + 6 = 3x – 1
QR: 2(7) + 6 =
QR = 20 units
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Example 4Find RT and SU.
Use Theorem 10-14 SV • VU = RV • VT
x • 4x = (x + 1) • 3xx
U
SR
Tx + 1 V 3x
4x 4x2 = 3x2 + 3xx2 = 3xx = 3
RT = (x + 1) + 3x RT = (3 + 1) + 3(3) RT = 13
SU = x + 4x SU = 3 + 4(3) SU = 15
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HomeworkSection 10-3
Page 667 – 668 (3 – 11, 15, 18 – 20)
Section 10-6Page 692 – 693 (3, 4, 13, 16)