12.3 arcs and chords
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12.3 Arcs and Chords
Geometry
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Objectives/Assignment
• Use properties of arcs of circles, as applied.
• Use properties of chords of circles.• Reminder Quiz Tomorrow!!!!!!
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Ex. 1: Finding Measures of Arcs
• Find the measure of each arc of R.
a. b. c.
MNMPN
PMN PR
M
N 80°
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Ex. 1: Finding Measures of Arcs
• Find the measure of each arc of R.
a. b. c. Solution:
is a minor arc, so m = mMRN = 80°
MNMPN
PMN PR
M
N 80°
MN
MN
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Ex. 1: Finding Measures of Arcs
• Find the measure of each arc of R.
a. b. c. Solution:
is a major arc, so m = 360° – 80° = 280°
MNMPN
PMN PR
M
N 80°
MPN
MPN
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Ex. 1: Finding Measures of Arcs
• Find the measure of each arc of R.
a. b. c. Solution:
is a semicircle, so m = 180°
MNMPN
PMN PR
M
N 80°
PMN
PMN
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Ex. 2: Finding Measures of Arcs
• Find the measure of each arc.
a. b. c. m = m + m =40° + 80° = 120°
GE
R
EF
G
H
GEFGFGE
GH
HE
40°
80°
110°
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Ex. 2: Finding Measures of Arcs
• Find the measure of each arc.
a. b. c. m = m + m =120° + 110° = 230°
GE
R
EF
G
H
GEFGF
EF
40°
80°
110°GEF
GE
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Ex. 2: Finding Measures of Arcs
• Find the measure of each arc.
a. b. c. m = 360° - m =360° - 230° = 130°
GE
R
EF
G
H
GEFGF
40°
80°
110°GF
GEF
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Theorem 12.6• In the same circle, or in
congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
ABBCif and only if
AB BC
CB
A
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Theorem 12.7
• If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
E
D
G
F
DG
GF
, DE EF
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Converse of Theorem 12.7
• If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
JK is a diameter of the circle.
J
L
K
M
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Ex. 4: Using Theorem 12.6
• You can use Theorem 10.4 to find m .
AD
• Because AD DC, and . So, m = m
AD
DC
AD DC
2x = x + 40 Substitute x = 40 Subtract x from each
side.
BA
C
2x°
(x + 40)°
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Finding the Center of a Circle
• Theorem 12.7 can be used to locate a circle’s 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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Finding the Center of a Circle
• Step 2: Draw the perpendicular bisector of each chord. These are the diameters.
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Finding the Center of a Circle
• Step 3: The perpendicular bisectors intersect at the circle’s center.
center
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• 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.
Theorem 12.9
F
G
E
B
A
C
D
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Ex. 7: Using Theorem 12.9
AB = 8; DE = 8, and CD = 5. Find CF.
58
8 F
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C
E
D
A
B
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Ex. 7: Using Theorem 12.9
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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8 F
G
C
E
D
A
B
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Ex. 7: Using Theorem 12.9
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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8 F
G
C
E
D
A
B
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Reminders:
• Quiz after 12.3 • Last day for seniors is this Friday, make
sure you return your books!
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• Homework: Finish the worksheet 12.3• Last day for seniors is this Friday, make sure you
return your books!
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