circles day 3 segments and other angles and more2013
TRANSCRIPT
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•Turn in CR # 3 on the bookshelf and work from Friday(two nice neat piles). Complete the drill
Drill 6/2/14
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Drill 6/2/14• Given: circle O, with A and B as
points of tangency and mBVA=32, find the
VA
B
O
measure of arcBCA.
C
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Find the measure of each arc
142 x
x2
103 x
x4
x3
EAe
DEdCDc
BCbABa
.
..
.. D
C
BA
E
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•Key terms–1) intercepted arc–2) secant–3) chord
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Objective•Students will prove theorems regarding angles in circles.
•Students will prove the segment theorems for circles.
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Inscribed Angle Theorem
•An angle inscribed in an arc has a measure equal to one-half the measure of the intercepted arc.
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Inscribed Angle Theorem
•m AVB= 1/2mAB
A
V B
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Inscribed Angle Corollary
•An angle inscribed in a half-circle is a right angle.
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Vertex On•mAV=180•m AVC=90
• mAV=m AVCV
A
C12
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Vertex On•mAV=x•m AVC= x
• mAV=m AVCV
A
C
12
12
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Vertex On•mABC=x•m AVC= x
• mAV=m AVCV
A
C
12
12
B
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Vertex In• (mAD+mBC)=• m AVD=• m BVC
V
A
C
12
B
D
O
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Vertex Out• (mBD-mAC)=• m BVD=
V
A
C
12 B
D
O
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IJ
FGHm
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M<QPR =________
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Congruent Tangents Theorem
•Tangents to a circle from the same external point have equal measure.
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Congruent Tangents Theorem
•BV=AV
A
V
B
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External Segment•BV and AV are external
AV
B
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Secant Proportion Theorem
• If two secants intersect outside of a circle, then the product of the lengths of one secant segment and its external segment equals____________ ________________________.
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Secant Proportion Theorem• If two secants intersect outside of a
circle, then the product of the lengths of one secant segment and its external segment equals____________
• ________________________the product of
the other secant and its external segment.
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Segment Proportion Theorem
AX XC=BX XD
A
X
B
C
D
Whole x External=Whole x External
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Secant-tangent Proportion Theorem
• If a secant and a tangent intersect outside of a circle, then the product of the length of the secant segment and its external segment equals __________________________________________________
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Secant-tangent Proportion Theorem• If a secant and a tangent
intersect outside of a circle, then the product of the length of the secant segment and its external segment equals __________________________________________________the length of the tangent segment squared.
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Secant-tangent Proportion Theorem
AX XC=(EX)2
A
X
E
C
Whole x External=Tangent2
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Intersecting Chords Theorem• If two chords intersect inside a
circle, then the product of the lengths of the segments of one chord equals ___________________________________________________________
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Intersecting Chords Theorem• If two chords intersect inside a
circle, then the product of the lengths of the segments of one chord equals ___________________________________________________________
The product of the lengths of the segments of the other chord.
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Intersecting Chords Theorem•AX CX=BX DX
A X
D
CB
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Secant Proportion Theorem
• If two secants intersect outside of a circle, then the product of the lengths of one secant segment and its external segment equals____________ ________________________.
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Secant-tangent Proportion Theorem
• If a secant and a tangent intersect outside of a circle, then the product of the length of the secant segment and its external segment equals __________________________________________________
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Intersecting Chords Theorem• If two chords intersect inside a
circle, then the product of the lengths of the segments of one chord equals ___________________________________________________________
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The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the
circle by
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A segment of a circle is a region bounded by an arc and its chord.
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In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle.
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Geom Drill Cont B and C are centers; E and D are points of
tangency
B C
DE
47x
10
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B and C are centers; E and D are points of tangency
•Find x
B C
DE
47x
10
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Find GB and GC
B C
DE
47
x
10
G
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Find GB and GC•GB=3 GC=10
B C
DE
47
x
10
G
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To find x, use pythagorus•32+102=x2
B C
DE
47
x
10
G
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To find x, use Pythagorus•X=√109
B C
DE
47
x
10
G
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Find the perimeter if the polygon has been
circumscribed
4 8
6
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BC is tangent to circle A. Find x
x8
x 4