•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