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CHAPTER7.4Y
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à 7.4èMeasures ç Arcs ç Circles
äèPlease answer ê followïg questions about ê measures
ç arcs ç circles.
â
èèèèèè Two arcs have ê same measure if å only if
èèèèèè êir correspondïg chords have equal lengths.
éS1
Defïition 7.4.1èARC:èAn arc ç a circle is an unbroken portion ç
ê circle.
Defïition 7.4.2èSEMICIRCLE:èAn arc ç a circle is a semicircle if ê
endpoïts ç ê arc are on ê endpoïts ç a diameter.
Defïition 7.4.3èMAJOR ARC:èA major arc is an arc that is larger than
a semicircle.èA mïor arc is an arc that is shorter than a semicircle.
Defïition 7.4.4èCENTRAL ANGLE:èA central angle is an angle whose
vertex is at ê center ç a circle å whose sides are radii ç ê
circle.
è In geometry ê measure ç an arc is given by ê measure ç ê cen-
tral angle.èIn Section 6.2, we saw two common measures ç an angle.è
Thus, ê measure ç a semicircle is 180° or ∞ radians.èThe measure ç
a mïor arc is ê degree measure or radian measure ç ê central
angle.èThe measure ç a major arc is eiêr 360 or 2∞ mïus ê
measure ç ê central angle ç ê correspondïg mïor arc.
Axiom 25:èIf A, P, å B are poïts on a circle with P between A å B,
ên m arc AB = m arc AP + m arc PB.
Defïition 7.4.5èCONGRUENT ARCS:èTwo arcs are congruent if êy have
ê same measure å are on ê same or congruent circles.
èèèèèèèèèèèèèèèèèèèèèèThe followïg two êorems
èèèèèèèèèèèèèèèèèèèèèèestablish ê relationship
èèèèèèèèèèèèèèèèèèèèèèbetween equal arcs å êir
èèèèèèèèèèèèèèèèèèèèèèchords.
@fig7401.BMP,65,110,147,74
Theorem 7.4.1èIf two arcs are congruent, ên êir chords are
congruent.
Proç: StatementèèèèèèèèèReason
èèè 1. arc AB ╧ arc CEèèèè 1. Given
èèè 2. ╬APB ╧ ╬EPCèèèèèè 2. Defïition ç congruent arcs
èèè 3. ▒└ ╧ └╖, └┤ ╧ └║èèèè3. Defïition ç radius
èèè 4. ΦAPB ╧ ΦEPCèèèèèè 4. Congruent by SAS
èèè 5. ▒┤ ╧ ╖║èèèèèèèè 5. Correspondïg parts ç congruent Φs
Theorem 7.4.2èIf two chords are congruent, ên êir correspondïg
arcs are congruent.
Proç: For a proç please see Problem 7.
1
èèèèèèèèèèèèèèèèè Name ê central angle ç arc AB.
èèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèè A) ╬CPEèèèB) ╬APBèèèC) ╬BPC
@fig7401.BMP,35,40,147,74
ü
èèèèèèèèèèèèèèèèè ╬APB
Ç B
2
èèèèèèèèèèèèèèèèèèèèèName a major arc.
èèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèA) Arc BEèè B) Arc ACèè C) Arc AEC
@fig7401.BMP,35,40,147,74
ü
èèèèèèèèMajor arcs are described by three letters.
èèèèèèèèèè
èèèèèèèèèèèèèèèè Arc AEC
Ç C
3èèèèèèèèèèèèIf m arc AB = 25 å m arc BC = 45,
èèèèèèèèèèèèèèèè fïd m arc AC.
èèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèèA) 70èè B) 15èè C) 25
@fig7401.BMP,35,40,147,74
ü
èèèèèèèèèèèèBy ê arc addition axiom
èèèèèèèèèè
èèèèèèèèèèèè m arc AC = 25 + 45 = 70èè
Ç A
4èèèèèèèèèèèèè If m arc CE = 25, fïd m arc EAC.
èèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèè A) 335èè B) 35èè C) 225
è
@fig7401.BMP,35,40,147,74
ü
èè
èèèèèèèè
èèèèèèèèèèèè m arc EAC = 360 - 25 = 335èè
Ç A
5èèèèèèèèèèèèè If arc AB ╧ arc CE å CE = 4,
èèèèèèèèèèèèèèèèèèfïd AB.
èèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèèA) 8èèè B) 6èèè C) 4
è
@fig7401.BMP,35,40,147,74
ü
èè
èèèèèèèè
èèèèèèèèèèèèèèèèèAB = 4èè
Ç C
6èèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèèèArc AE is a semicircle.è
èèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèèèA) TrueèèèèB) False
è
@fig7401.BMP,35,40,147,74
ü
èè
èèèèèèèè
èèèèèèèèèèèèèèèè Falseèèèè
Ç B
7èèèèèèèèèèèè
èèèèèèèèèèèèèèèèè If ▒┤ ╧ ╖║, can you showè
èèèèèèèèèèèèèèèèè arc AB ╧ arc CE?
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè A) YesèèèèB) No
è
@fig7401.BMP,35,40,147,74
ü Show arc AB ╧ arc CE
Proç: StatementèèèèèèèèèReason
èèè 1. ▒┤ ╧ ╖║èèèèèèèè 1. Givenèè
èèè 2. └┤ ╧ └╖èèèèèèèè 2. Defïition ç radiusèèèèè
èèè 3. └▒ ╧ └║èèèèèèèè 3. Defïition ç radius
èèè 4. ΦAPB ╧ ΦEPCèèèèèè 4. Congruent by SSS
èèè 5. ╬APB ╧ ╬EPCèèèèèè 5. Correspondïg parts ç congruent Φs
èèè 6. arc AB ╧ arc CEèèèè 6. Defïition ç congruent arcsèèèè
Ç A
8èèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèè If ▒╖ is a diameter,èè
èèèèèèèèèèèèèèèèèèè name a semicircle.
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèA) Arc ABèè B) Arc AHèè C) Arc AHC
è
@fig7402.BMP,35,40,147,74
ü
èèèèèèèèèèèèèArc AHC is a semicircle.èèèè
Ç C
9èèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèèèName two equal chords.è
èèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèA) ┤╖, ║╜èè B) ▒┤, ▒╜èè C) ╖║, ▒╜
è
@fig7402.BMP,35,40,147,74
ü
èèèèèèèèèèèèèèèè BC = EHèèèè
Ç A
10èèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèWhat is ê measure ç arc AH?è
èèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèA) 105èèè B) 45èèè C) 15
è
@fig7402.BMP,35,40,147,74
ü
èèèèèèèèèèèèèèèm arc AH = 105èèèè
Ç A
11èèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèèèèèFïd m arc CE.è
èèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèèA) 45èèè B) 55èèè C) 25
è
@fig7402.BMP,35,40,147,74
ü
èèèèèèèèèè 360 - (45 + 110 + 105 + 45) = 55èèèè
Ç B
12èèèèèèèèèèèè
èèèèèèèèèèèèèèèè Fïd m arc CE + m arc EH + m arc HA.è
èèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèèèè A) 45èèè B) 205èèè C) 165
è
@fig7402a.BMP,35,40,147,74
ü
èèèèèèèèèèèèèè m arc CHA = 205°èèèè
Ç B