Circles day 3 segments and other angles and more2013
42
•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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Transcript of Circles day 3 segments and other angles and more2013
•Turn in CR # 3 on the bookshelf and work from Friday(two nice neat piles). Complete the drill
Drill 6/2/14
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
Find the measure of each arc
142 x
x2
103 x
x4
x3
EAe
DEdCDc
BCbABa
.
..
.. D
C
BA
E
•Key terms–1) intercepted arc–2) secant–3) chord
Objective•Students will prove theorems regarding angles in circles.
•Students will prove the segment theorems for circles.
Inscribed Angle Theorem
•An angle inscribed in an arc has a measure equal to one-half the measure of the intercepted arc.
Inscribed Angle Theorem
•m AVB= 1/2mAB
A
V B
Inscribed Angle Corollary
•An angle inscribed in a half-circle is a right angle.
Vertex On•mAV=180•m AVC=90
• mAV=m AVCV
A
C12
Vertex On•mAV=x•m AVC= x
• mAV=m AVCV
A
C
12
12
Vertex On•mABC=x•m AVC= x
• mAV=m AVCV
A
C
12
12
B
Vertex In• (mAD+mBC)=• m AVD=• m BVC
V
A
C
12
B
D
O
Vertex Out• (mBD-mAC)=• m BVD=
V
A
C
12 B
D
O
IJ
FGHm
M<QPR =________
Congruent Tangents Theorem
•Tangents to a circle from the same external point have equal measure.
Congruent Tangents Theorem
•BV=AV
A
V
B
External Segment•BV and AV are external
AV
B
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____________ ________________________.
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.
Segment Proportion Theorem
AX XC=BX XD
A
X
B
C
D
Whole x External=Whole x External
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 __________________________________________________
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.
Secant-tangent Proportion Theorem
AX XC=(EX)2
A
X
E
C
Whole x External=Tangent2
Intersecting Chords Theorem• If two chords intersect inside a
circle, then the product of the lengths of the segments of one chord equals ___________________________________________________________
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.
Intersecting Chords Theorem•AX CX=BX DX
A X
D
CB
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____________ ________________________.
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 __________________________________________________
Intersecting Chords Theorem• If two chords intersect inside a
circle, then the product of the lengths of the segments of one chord equals ___________________________________________________________
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
A segment of a circle is a region bounded by an arc and its chord.
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.
Geom Drill Cont B and C are centers; E and D are points of
tangency
B C
DE
47x
10
B and C are centers; E and D are points of tangency
•Find x
B C
DE
47x
10
Find GB and GC
B C
DE
47
x
10
G
Find GB and GC•GB=3 GC=10
B C
DE
47
x
10
G
To find x, use pythagorus•32+102=x2
B C
DE
47
x
10
G
To find x, use Pythagorus•X=√109
B C
DE
47
x
10
G
Find the perimeter if the polygon has been
circumscribed
4 8
6
BC is tangent to circle A. Find x
x8
x 4
