Geometry
10"1
-
Use ProBerties of Tangents
Learning Target: By the end of today's lesson we will be able to successfully use properties of a tangent to a circle,
lllustratisn
Definition
Vocabulary
A circle is the set of all points in a plane Circle
that are €qqiAis$oot from a given point.
Oc
The center of a circle is the Center
ooiot thl;.|" .* A
Radius
from which all Points of .quidistant.
from the of a circle to any
ce,n{e?
point on the circle is a radius.
Chord
A diameter is
Diameter
Secant
Tangent
1)
contains circle.
a chorA
the
Cen{c,f
a
na in 3.
A secant is Ir intersects a circle
A tangent is
a I inc
exactly I
point.
tnat of the
that points.
in the plane of a circle that intersects the circle in
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of circle
al
BC
r qA irrs
bl
EA
Seca.r{
cl
DE
*""'3.''*
C.
21
Use the diagrarn to find the given lengths. a) Radius of circle A
c)
b)
Diarneter of circle A
d) Diameter of circle B
Radius of circle B
@ 3) Tell how
"*"Waveandtt",H ',{g 3
nnany common
t*/
--
o
,1,
ln a plane, a line is tangent to a circle if and only if the line circle at its endpoint on the circle.
4,
ln each diagram,
, ZhL
5t+
looa5Tc =L1L ._<\
R?
c,.\or to .
tlnAiu
<
f-*"
i'-t'h 1 U t\ \ -/ \."
(py't\"3o".T
rf " 8*
*i,f);l'-@
Gtt=L-tlnr'W
ln the diagram, I is a point of tangency. Find the radius r of circle C.
F
OHIT
F2+ 1-lt=
*';::)
,/---\-----\22 lc" --\ /]
is
3x+5 = ?7
3X=il
" @
OH IT
(rt
3z)z
rz+ ?\3L = rt+
[*rt
8)
E6*=
3 t?o
a 6*r+ lo:'4
tltl
=
/\,ftr / I p'
are Con$rrlcnt
tangent to
'*-i:i;w@ f+
y'a 5ia1 = r'+13. + 2{ol 51at= 93r+tr*o1 j5a3 = 98r
tangent to circle C at R and OS circle C at 5. Find the value of x.
lf+ t['=]a
ln the diagram, K is a point of tangency. Find the radius r of circle L.
-2
is
I
6)
(r+qq)"
Tangent segments from a common external Point
QR
P. 'rAi
b)
lo"+ 2qa
-,t
p..
RS ir a radius of circle R. ls SZ tangent to circle
a)
5)
is
\\
nS
6tr E
\=Y-
J \"
l-d" ,=_rT
ir tangent to circle C at 5 and RZ is tangent to C at f. Find the value of x.
circle
Xt= rl i
,/;" =6'{
lozY
Geometry
10.2
- Find Arc Measures
LearningTarget: Bytheendoftoday'slessonwewillbeabletosuccessfullyuseanglemeasurestofindarcmeasures. lllustrations
Definitions
Vocabula
A central angle of a circle is an angle
Central angle
whose vertex is the
ccnl.r
of the circle.
Minor arc
Major arc
Part of a circle measuring less than
lt
o"
Part of a circle measuring between
Ito'
and 3Go'.
A semicircle is an arc with endpoints
Semicircle
that are the endpoints of
a
Ji^*.{.t The measure of a minor arc is the
Measure of a minor arc
measure of
its C-t"tr"*
|
The measure of a major arc is the
Measure of a major arc
36oo
difference between
and
the measure of the related Yrri
Congruent circles
nor
4fc
Two circles are congruent circles if
they have the same
rq,lius
3oc Two arcs are congruent arcs if theY have the
Congruent arcs
same fn@SufL
and they are arcs of the same circle or
Conqrucrrl of__-------J-
circles.
ftr
Ab.
;n
MEASURITUG ARES:
The expression srA& is read as "the measure of arc ,
r€ nl€osUre of a
minor arc is the measure of
The measure of the entire circle is
1)
is
36oo
mA&
l8D
$F
b,.
36o'
* Eo*
_
rn*Dg = 31so
andthemeasureoftherelatedminorarc.
o
Find the measure of each arc of circle C, where
a.
{ {:8 i .^ ,o/ln4 ^ .jtr_---*_/ J t,,
its Cc,n*ra"l angle.
Themeasureofarnajorarcisthedifferencebetween The measure of a semicircle
/--*H
AB".
Are
DF
c.
is a diameter.
&EF
ARC ADDITION POSTULATE:
The measure of an arc formed by measures of the two
arcs.
two Odi"o"rt
arcs is the
Stah
of the
t_
or*,8"S-:nr AB
,
_l +rlr Bc
You join a new bank and divide your money several ways, as shown in the circle graph at the right. Find the indicated
arc measures.
b)
a) *n6fi 65"t lo5"-
c) n;lt*g
n:S*S
l+oo + Go'
d)
o=
"*;iES.
55o+l*o"
fr"i'l 3) Tell whether the given arcs are congruent. Exploin why or why not. -"-!
a) S*
nnd fr5
rl
r'fl\
\
i-
\_l \rlc.s Sawrc-
and HJ s
* No clra\eS .F,
6fC. noh :
qe-.{rorl (s ln the diagram, is
F*
/f,\
trnc464ft-
4l
SS
"Jl
/r..\lt."\ j]r------{--'-l,
-
b) AS nnd
ffii : iil?
rxplain why or why not'
Y..
srrnc. cr".\rol (s sarqg rqAii
.rl"-14 l \\_-,"/--i\ Jh /L\ t-t I !-iJ
\l
\&s t
3at.tc. cerA.-\ s.^n^e
fa{i i
(
10.3
Geometry
* Applv Froperties of ehords
Learning Target: By the end of today's lesson we will be able to successfully use relationships of arcs and chords in
a
circle.
ln the same circle, or in congruent circle, two minor arcs are
their corresponding
1)
ln the diagra m,
ef*"\s
I
c-ongrucntif
and only if
are congruent.
OA= OD, nC = EF,and urii =
125. rlnO
*gg.
lLa /-2-\
{,{\
.s
mA=l}s"
'( \ "\/ \/
of another chord,
lf one chord is a then the first chord is a
Ji^"r..1.tf QS
is a perpendicular bisector
then
8S
of TR,
is a diameter of the circle.
/N
lf a diameter of a circle is then the diameter
bi
s.-k
17f
lf EG then
2l
Use the diagram of OE to find the length
{
'f-*dt 1. \fT
the chord and its arc.
BD . e(DF)
BD' e(bl flpe lL
is a diamete
HD =
nr
r
and
EG
u*6
L DF ,
ano ^^ 6D
=
GF
of BD. How do you know?
is J- 't" J.i"^"Lt to i+ is q. I bis..l'r
C!,."d
ln the same circle, or in congruent circles, two chords
are
Co,nqrutnl
if and only if
they are equidistant from the center. if and only
if EF = E6t
r-;
3)
ln the diagram of OF, AB = CD = 12. Find _\-
1 -r
A./
:i-
/
irx-
lr;*i
1'-' -f1-: '\.1,i-'-
'r| n ii
Jx-3 = 3* 4x=
J
- --."t --- --,':
4l
lf
r*Ti/ O
:
EF.
\ x-"
j
rt
EF=
7x
?F=
3ta)
EF=
L21o, find mS$"
f _
rd.-"--"Fri
i,."f1
lJ,',' n\V *--*r
s)
Finc{ the rHs{tsilr*s
*f frF"SF" nnd #S"
t*=Xo-* 5>t= $u
x--
6: = $(rul = L.lt
16
gC =Grl"
G - Lq" CE
= le,to
L
Name
Geometry
Date
10.1 - 10.3 Review Worksheet Show allwork for full creditl
Mat,rh the r:'siatinn witlr fh,e tsrm thfft hest descrilre,s 'it.
1. rl E A. {.rltr-:r z. f'I{ (i B. {:h':'ri{ g. ir"i,.) D *. Di"rmetpr 4. -1.,j B D. RrJtrs s. *l A E. F*int rf tfl,rlgsxrcy S, .,{,} C F. {-.r:n:rnrir extsrnal lnnge'n,t ?. ;F H G. t]*rrur:cn intrrnrl tirnp*ni 8. /lf F H. S*crrrl Draw 2 circles with the given number of tangents. 11) 1 Tangent
10) 2 Tangents
the cliag'rarn. ggls,a: r,*diu* s{' ,t'S. B,etertmin* tsh*'ther gA ls tFttgetl* to l+F. Explat-n V$rrr reassrrlng. *n,
rz.
t-F"-*- 3tr\"={
i+=rf^\ t+ra=rE *
\_*J
{'""i
[,n
13.
l
)S
=
"s
-A 9+3t +i
rr5
El
tlle rliagranr, ,4Fis tang+nt t* rI'#*t p*int F. Fittd th*
I4. A
D
f'tf
**L' =1
r'adit"lp
rsf {if.
(z+.lt'- (r + z)z 12+ ll, : r2+ {r $ { t6=tlrltl 12e{r
3:r
;g i* tfiHg,*lrlt to ,*{ at rfailsl Jilf i* tnn,g,ent ta i},{ a* ff 15.
3X -- Y+ la
\* : lf.
E\
z
Find t}re ualus llf x.
+ *tEJ
,{Fmrtel FErtn* d,lxnlr*t*rs *f'' .:,'S- s,*t*r*eir:e whatfusr il*pr a:rd; fri.*J6l/' sr{."! s r ..sd}JItJsJ, tlrre glv*metrg-l
.l;, bi
rs.
lfi. i
.,,ri:
qf
17.
l J.-,ii'
Se,micircl..
19.
f,ii,,ti
ry1a16r qfC
ir"r lloo :oo,r'1---'\ \. i '.. \n 7n'f? ..,[ :Llir,
[n il#, F]S anal ltr:* are sliarftete,r#. Find the i
nqli$st,*tl 'n1*ssur€,,
2{r. ,ol:iy 22, ilr,l.'l:P
@
tl
@J
4'J,
Tsll uuhether FE
24' jt'-'-.. o{,
ta
\
,l
,
8o\ i ',/
/
7oo
";;.n =.fg'.
Exp{aiw.
,r--
Yes
ia.r-'i, ' v.'t[-.o1 o.6t"s *-Ir," *-D
t'l
\-
o'rc,
So
co
frsr6.
ths *h+rql length. 26. All
\es il
i\''--/'I5:j^"g'n* c ir.\.s ..r" $* scrr\e --l kro.^rse, +t ^, hnvc, lk f*ur<- r^Alus orrd ttr.- hovs \\tse.tx, ct .\,o't *S1". '
""
nlr,itr,t
.--J
Firrd:
2?" F.{;
D
:f
"d;?t r==-=c\
l iiGl t ll,(^ .u i
I
\,.r,Jl \. : ;\:L--l ,4h
r
\-r,tj
)
i j
a
1i \.-I---
\rc
/
ths ar* lengttr.
Fins*
zs. u,11' 0
=@
lo,t-J I ,t'' \
x.{'"' (, f
.,.',
^
j||ii-.J
re. *u'#' =
s{
\
o /t'..
!
/i-
7
:
E
\
35[r,-.1*'-lis" \ix*Er*" .]
-
ni \\,"1-4an"
|
-v
\."
R'\--l
f.!-J
$'v
Tell us'h*ther Ssis a disimeter qf C'S.
30.
#
Y.. Li.".lt
lf neit* expl*in
e.\norJ
{o
N" cds {r.
.h.;
i" h^\f "^A is pe.p-..di..*\or
why-
T-R.
l,''{o *wo
unel*J fr rl S
Tell whether the measures are equal. 44
ra'
J,f'
,1tllr:1
33. mSf
Ir,\
nnd ln.&S ri
nd
Ao
Firrr{ the given rn€'Hsrtr€.
34. sgr
n"l I
=G
gs. ,r.i,r, =t;55-\
,eTe.+t :'.r'J #t** ln \;'
Tsll vqhether ths length* are
/
e*1rrel.
36. f'{l snei f;F
.,{il;rnd fil1 fi
,41* \ J
ti
1,
l.''X"*cl I " -'+, l{j rr t; --'--"
F- r+q\
Find the qliven rffiB€suti*"
3n.
,g'[T[
3e. i1{1 rEf r
ff r-.fr-, /i l--r \-r *-i\ ff
-F* oir"1l;--,\z ,"19
llrrl 1
-J
-'
..1
|
ll_1;J" + i,,i7 \14 \. lr #\--'I
I
l
-: -t'4 J s-\-\]trY. F:ind
the u*lu* sf x.
4t1.
f {ay+ 'l1n-}, . I r,r4J r' *"--'i i
s\. iix- r''i j \_ ,--lE
6*+t = 3x -€ (" s zl,
1:X
B
41.
tx-l
'3t{-G
Geometry
Name f 0.2
Date
Hour
Find
the measure of each arc of circle
1)
i.;'E
zt
-
1,0.3 Review
Worksheet
Show all work for full credit! J, where segment KM is a diameter.
3) ffi{iE
LrrI'@
lSor ldentify the given arc as a major arc, minor arc, or semicircle. Then, find the measure of the arc.
5):{"li@
4t::tt?G
rni^ o r
rn'irrrec
7)
G) ilTi tiE-o.;{
sijr.f]
Serni
s) 'fc$ F;1
'iJa
rbLl;;'
i{
Qb". t'
ili:f,-lB*, r v j/ f
e) #t' [sb1
*\\*
*\\".
circt<-
tnr"nor
Several Students were recently asked about their favorite color. The results are shown in the graph. Find the indicated grE*fi rmeasures : af :,
ro1
t2l
fafi
,*rffiB
'y
o*Frff@
ffi rr1 *'sfg h'+5"\
rrrli
?-
*Fsf 360 -rf
r?* **-
'"
blu* T laclt
ldentify the given arc as a major arc, minor arc, or semicircle. Then, find the measure of the arc. J{- '
L4t;trs.El 1s)'r'Gl 5sr"ri
mi
cir"l.-
'.^..-.11
L4 fL{\l1fl wrin"'r
tr)
N
16)
nor
fnc6oc
le);ir @
*i:,f,@
rneioc
Sconi"L'-\t-
Tell whether the highlighted arcs are congruent. Explain why or why not.
20)
/ffi t.s j vqrga!(s t {1i,
Fgdr
arc --
21)
F
r.,a"m r',Y.J "\-"1"} Lr __v.s
'
ralirrS c,,.J c<.rtrl {. icw.c.
)eS tem. rcl..^9 a^A .r^L"\
(
Desffibe in words what you can conclude about the diagranr. z3)
p is the center of the circle. use the given information to find XY.
261
,f
!:"
:*
{i, -!
{'{'" ::::
'f
r
?,5
Find the measure of arc MN. 28)
f
,/ I
2-
-1'
----*i .-t\-
,1
:ll
-.---
f
*.'r'. :J I ! l ,J'< ./, -! -.- ,, \*r \.. y'
.rt,'-ir ; i
a <3
\
)
.'
};:-a--r
I
a
,r,
30) rS
360
,r{*\
I
t'./i-,'
| ^r ,"f," '"
"{
._
rr
]
i
/ t" )
F
65
i
t6s
1lv '\J \rr./
--__ j:_-n
-z{ o lzo
31)
-.-jl.-rr ,/\ 1.:\
ti"ii*;,-
l;1*
321
y'l::;":
32+ rl2= c.z
1*lb--c-
N,\
nl
SE '---.-/----'
s
N=
14N =
60o
o_ lD-b--i=/.
\
9o"
l5o
o
-
360
l5D
l*\rrl
: loo"
Ll o
\ to,' '\,,
io ./ -r'r
t
\\1
'
-#-I*-"-.
i.r-
,t\*-
o
;r4;1 = loL
t!
Mf
3(o
It?
1t'
141
al
xx
lor
t?20
j*i
'"' 6b"
\OLL*f
3s)
34)
33)
I wi - -*--"t\,,
MN: IoS
l3e
rl
-zLtl
n
?60
ltN
: {8'
3Lo
lL 2q4 .le9 3?
Mt'l c 3f
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