Post on 15-Feb-2016
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Unit 4:Arcs and Chords
Keystone Geometry
Measures of an ArcMeasures of an Arc:1. The measure of a minor arc is the measure of its central angle. 2. The measure of a major arc is 360 - (measure of its minor arc). 3. The measure of any semicircle is 180.
Adjacent Arcs:Arcs in a circle with exactly one point in common.
A
B
C
T
P
List: Major Arcs Minor Arcs Semicircles Adjacent Arcs
Example: Find the measure of each arc.4x + 3x + (3x +10) + 2x + (2x-14) = 360°
14x – 4 = 360° so 14x = 364x = 26°
4x = 4(26) = 104°
3x = 3(26) = 78°
3x +10 = 3(26) +10= 88°
2x = 2(26) = 52°
2x – 14 = 2(26) – 14 = 38°
3x+10
2x-14
2x4x
3x
B
D
C
E
A
4
Theorem• In the same circle or in congruent circles, two
minor arcs are congruent if and only if their central angles are congruent.
M
N
R
S
O
5
Arc Addition Postulate• The measure of the arc formed by two
adjacent arcs is the sum of the measures of these two arcs.
H
J G
K
F
6
Example• In circle J, find the measures
of the angle or arc named with the given information:
• Find:
H
J G
K
F
7
In Circle C, find the measure of each arc or angle named.• Given: SP is a diameter of
the circle. Arc ST = 80 and Arc QP=60.
• Find:
T
C
P
S
Q
8
Theorem #1:In a circle, if two chords are congruent then their corresponding minor arcs are congruent.
E
A B
C D
Example:
9
Theorem #2:In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.
E
D
A
C
BExample: If AB = 5 cm, find AE.
10
Theorem #3:In a circle, two chords are congruent if and only if they are equidistant from the center.
O
A B
C
DF
E
Example: If AB = 5 cm, find CD.
Since AB = CD, CD = 5 cm.
CD AB iff OF OE
11
Try Some:• Draw a circle with a chord that is 15 inches long and 8
inches from the center of the circle. • Draw a radius so that it forms a right triangle. • How could you find the length of the radius?
8cm
15cm
O
A BD
∆ODB is a right triangle and OD bisects AB
2 2 2
2 2 2
AB 15DB= = =7.5 cm2 2
OD=8 cmOB =OD +DBOB =8 +(7.5) =64+56.25=120.25OB= 120.25 11 cm
Solution:
x
12
Try Some Sketches:• Draw a circle with a diameter that is 20 cm long.
• Draw another chord (parallel to the diameter) that is 14cm long. • Find the distance from the small chord to the center of Circle O.
Solution:
∆EOB is a right triangle. OB (radius) = 10 cm2 2 2
2 2 2
2
10 7
100 49 51
51
OB OE EB
X
X
X
10 cm10 cm
20cm O
A B
DC
7.1 cm
14 cm
Ex