Transcript 10-2 Measuring Angles and Arcs

Arcs and Central Angles
Section 11.3
Goal
• Use properties of arcs of circles.
Key Vocabulary
•
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Central angle
Arc
Minor Arc
Major Arc
Semicircle
Congruent Circle
Congruent Arcs
Adjacent Arcs
Arc Length
Postulates
• 16 Arc Addition Postulate
Central Angle
Definition: An angle whose vertex lies on the center of the circle.
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central
Angle
(of a circle)
Central Angle
• Central angle- Its sides contain two radii of
the circle.
A
ACB is a central
angle
C
B
Sum of Central Angles
• The sum of the central angles of a circle = 360o
– as long as they don’t overlap.
Example 1
Example 1b:
Refer to ⊙T.
Assume RV is a
diameter.
Find the m∠QTR.
Example 1b:
∠QTR and ∠QTV
m∠QTR + m∠QTV = 180
m∠QTR + 20x = 180
form a linear pair.
Linear pairs are supplementary.
Substitution
m∠QTR + 20(7) = 180
m∠QTR + 140 = 180
m∠QTR = 40
Answer: 40
Simplify.
Subtract 140 from
each side.
Your Turn:
Refer to ⊙Z. Assume AD and BE are diameters.
a. Find m∠CZD
Answer: 65
b. Find m∠BZC
Answer: 40
K
L
O
Centre
An arc is the distance between any two points on the
circumference of a circle. Symbol:
⏜
Arcs
Every central angle cuts the circle into two arcs
The smaller arc is called the Minor
Arc. The MINOR ARC is always less
than 180°. It is named by only two
letters with an arc over them as in our
example,
.
The Minor Arc
The larger arc is called the Major
Arc. The MAJOR ARC is always more
than 180°. It is named by three letters
with an arc over them as in our
example,
.
The Major Arc
Minor Arc KL
K
An arc divides
the circle into
two parts: the
smaller arc is
called the minor
arc, the larger
one is called the
major arc.
L
O
Centre
Major Arc KYL
Y
Minor arcs are named by their endpoints.
Major arcs are named by their endpoints and
another point on the arc that lies between the
endpoints.
Semicircle
The Semicircle
(Major Arc = Minor Arc) : The
measure of the semicircle is 180°. SEMICIRCLES are
congruent arcs formed when the diameter of a circle
separates the circles into two arcs.
S
Semicircle
Arc DSE DSE
Diameter
E
D
O
Centre
R
Semicircle
Arc DREDRE
 Half of a circle is called a semicircle.
 A semicircle is also an arc of the circle.
 A semicircle is named with 3 letters, same as a major arc.
NAME THE ARC
ARCS
Arcs : The part or portion on the circle from some point B to C
B
is called an arc.
Example:
Semicircle:
Example:
C
BC
A
An arc that is equal to 180°.
ABC
A
B
O
C
Minor Arc & Major Arc
Minor Arc : A minor arc is an arc that is less than 180°
A minor arc is named using its endpoints with an “arc” above.
A
Example:
AB
Major Arc:
A major arc is an arc that is
greater than 180°.
B
A
A major arc is named using its
endpoints along with another
point on the arc (in order).
O
C
Example: ABC
B
Example: ARCS
Identify a minor arc, a major arc, and a semicircle, given that
is a diameter.
Minor Arc:
CD
DE, EC, CF , DF
E
D
Major Arc: CEF , EDC, DFE, FCD
A
C
F
Semicircle:
CED, CFD, EDF , ECF
True or False
The name of the orange
arc below is…..
True or False
FALSE…it is
Semicircle - named
using three points on the
arc; endpoints listed first
and last.
True or False
The name of the orange
arc below is…..
True or False
FALSE. It is…..
Minor arc - named
using the two
endpoint letters.
True or False
The name of the orange
arc below is…..
True or False
FALSE
Major arc - named
using three points on
the arc with endpoints
listed first and last.
Arc Measure
The measure of the an arc is equal to the measure of the central
angle.
An arc is measured in degrees, the same as an angle.
Arc
Y
Central Angle
O
110
Z
Definition of Arc Measure
The measure of a minor arc is the
measure of its central angle.
Central Angle = Minor Arc
The measure of
a major arc is
360° minus the
measure of its
central angle.
Example 2
Name the red arc and identify the type of arc. Then
find its measure.
a.
b.
SOLUTION
a. DF is a minor arc. Its measure is 40°.
b. LMN is a major arc. Its measure is 360° – 110° = 250°.
Example 3
Find the measure of GEF .
SOLUTION
mGEF = mGH + mHE + mEF
=
40° + 80° + 110°
= 230°
Find the arc measures
m AB = 80 
B
m DE = 45 
D
m AF = 45 
m DF = 180 
55 
80
A
45 
m BF = 125 
45
C
E
m BD = 55 
m DFB = 305 
F
m FE = 135 
Example 4a
Answer:
Example 4b
Example 4b
Answer:
Example 4c
Answer:
Your Turn:
A.
B.
C.
D.
Your Turn:
A.
B.
C.
D.
Your Turn:
A.
B.
C.
D.
Congruent Arcs
Within a circle or congruent circles,
congruent arcs are two arcs that have the
same measure. In the figure ST  UV.
Example 5
Identify Congruent Arcs
Find the measures of the blue arcs. Are the arcs
congruent?
a.
b.
SOLUTION
a. Notice that AB and DC are in the same circle.
Because = mAB = mDC = 45°, AB  DC .
b. Notice that XY and ZW are not in the same circle or
in congruent circles. Therefore, although
mXY = mZW = 65°, XY  ZW.
Checkpoint
Identify Congruent Arcs
Find the measures of the arcs. Are the arcs congruent?
1. BC and EF
ANSWER
mBC = 58°; mEF = 58°; yes
2. BC and CD
ANSWER
mBC = 58°; mCD = 72°; no
3. CD and DE
ANSWER mCD = 72°; mDE = 72°; yes
4. BFE and CBF
ANSWER
mBFE = 158°; mCBF = 158°; yes
Example 6a
mLPK = 0.21(360)
= 75.6
Answer:
Find 21% of 360.
Simplify.
Example 6b
Example 6b
Sum of arc in circle is 360.
Substitution
Simplify.
Simplify.
Answer:
Your Turn:
A. 124.3
B. 140.4
C. 155.6
D. 165.9
Your Turn:
A. 273.6
B. 240.5
C. 215.7
D. 201.4
Adjacent Arcs
Adjacent arcs are arcs of the same circle that
intersect at exactly one point (share an endpoint).
RS and ST are adjacent arcs.
Postulate 16 Arc Addition Postulate
• The measure of an arc formed by two
adjacent arcs is the sum of the measures of
the 2 arcs.
m CA + m DC = 72 
A
m CA = 40 
B
C
m DC = 32 
D
m DA = 72 
Example 7a
• Find m ABD
m CA + m DC = m AD = m ABD
4x + 7 + 2x + 5 = 8x
6x + 12 = 8x
A
B
m AC = 4x + 7 
8x 
C
m CD = 2x + 5 
D
m ABD = 8(6)
m ABD = 48 
12 = 2x
6=x
Example 7b
Line segments AC and BE are diameters of ⊙F.
Find mBD.
mBC = 97.4
Vert. s Thm.
mCFD = 180 – (97.4 + 52)
= 30.6
∆ Sum Thm.
mCD = 30.6
mBD = mBC + mCD
= 97.4 + 30.6
= 128
mCFD = 30.6
Arc Add. Post.
Substitute.
Simplify.
Your Turn:
Find each measure.
mJKL
mKPL = 180° – (40 + 25)°
mKL = 115°
mJKL = mJK + mKL
= 25° + 115°
= 140°
Arc Add. Post.
Substitute.
Simplify.
Your Turn:
Find each measure.
mLJN
mLJN = 360° – (40 + 25)°
= 295°
Arc Length
• Another way to measure an arc is by its length.
• The arc length is different from the degree measure of an
arc. Suppose a circle was made of string. The length of
the arc would be the linear distance of that piece of string
representing the arc.
• An arc is part of a circle, so its length is part of the
circumference. We use proportions to solve for the arc
length, l.
degree measure of arc = arc length
degree measure of
circumference
Arc Length
• Arc length is a part of the circumference of a
circle.
Arc length of AB mAB


2 r
360
r
B
x˚
ℓ
A
C
OR
mAB
Arc Length of AB 
 2 r

360
x

2 r
360
Example 8
Arc Length Equation
Substitution
cm
Answer:
Use a calculator.
Example 9
Arc Length Equation
Substitution
Use a calculator.
Answer:
Your Turn:
A. 3.56 cm
B. 3.77 cm
C. 3.98 cm
D. 4.21 cm
Example 10
Find the length of the red arc.
a.
b.
c.
SOLUTION
a.
50°
· 2(5) ≈ 4.36 centimeters
Arc length of AB =
360°
b.
50°
· 2(7) ≈ 6.11 centimeters
Arc length of CD =
360°
c.
98°
· 2(7) ≈ 11.97 centimeters
Arc length of EF =
360°
Checkpoint
Find Arc Lengths
Find the length of the red arc. Round your answer to
the nearest hundredth.
5.
ANSWER
4.19 in.
ANSWER
12.57 ft
ANSWER
9.42 cm
6.
7.
Example 11
Arc Length Equation
Substitution
Use a calculator.
Answer:
Your Turn:
A. 36.56 cm
B. 37.79 cm
C. 38.61 cm
D. 40.21 cm
Your Turn:
A. 32.99 cm
B. 33.59 cm
C. 33.89 cm
D. 34.61 cm
Assignment
• Pg. 604 – 607; #1 – 57 odd.