Download 12-2 - Ithaca Public Schools

Name_____________________________________ Class____________________________ Date ________________
Lesson 12-2
Lesson Objectives
1 Use congruent chords, arcs, and
central angles
2 Recognize properties of lines through
the center of a circle
Chords and Arcs
NAEP 2005 Strand: Geometry
Topic: Relationships Among Geometric Figures
Local Standards: ____________________________________
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Vocabulary and Key Concepts.
Theorem 12-4
Within a circle or in congruent circles
(1) Congruent central angles have
(2) Congruent chords have
(3) Congruent arcs have
chords.
arcs.
central angles.
Theorem 12-5
Within a circle or in congruent circles
(1) Chords equidistant from the center are
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(2) Congruent chords are
.
from the center.
Theorem 12-6
In a circle, a diameter that is perpendicular to a chord bisects the
and its
.
Theorem 12-7
In a circle, a diameter that bisects a chord (that is not a diameter) is
to the chord.
Theorem 12-8
In a circle, the perpendicular bisector of a chord contains the
of the circle.
A chord is
P
Q
O
Daily Notetaking Guide
Geometry Lesson 12-2
231
Name_____________________________________ Class____________________________ Date________________
Examples.
A
1 Using Theorem 12-4 In the diagram, radius OX bisects AOB.
What can you conclude?
AOX by the definition of an angle bisector.
AX because
central angles have
X
O
chords.
because
chords have
arcs.
B
2 Using Theorem 12-5 Find AB.
QS QR RS
Segment Addition Postulate
QS Substitute.
QS Simplify.
AB Chords that are equidistant from the center of a circle
are congruent.
AB Substitute
7
The distance from the center of O to PQ is measured along a
perpendicular line.
PM (
2
)
P
r
O
Use the Pythagorean Theorem.
2
Substitute.
Simplify.
r
232
M
Substitute.
r2 The radius of O is
S
A diameter that is perpendicular
to a chord bisects the chord.
OP 2 PM 2 OM 2
r2 7
Q
The distance from O to PQ is 15 in., and PQ 16 in.
Find the radius of O.
PQ
R
for QS.
3 Using Diameters and Chords P and Q are points on O.
PM 4
C
4
A
Q
Find the square root of each side.
in.
Geometry Lesson 12-2
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B
Daily Notetaking Guide
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0
AX Name_____________________________________ Class____________________________ Date ________________
Quick Check.
1. If you are given that BC DF in the circles, what can you conclude?
B
O
D
P
C
F
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O P
2. Find the value of x in the circle at the right.
18
18
16
x
36
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3. Use the circle at the right.
a. Find the length of the chord to the nearest unit.
6.8
4
x
b. Find the distance from the midpoint of the chord to the midpoint of its
minor arc.
Daily Notetaking Guide
Geometry Lesson 12-2
233