'Geometry
10.6
- Find Seement Lengths in Circles
LearningTarget: Bytheendoftoday'slessonwewillbeabletosuccessfullyfindsegmentlengthsincircles.
When two chords intersect in the
i Segments of a Chord
",{".
io.
chord is divided
of a circle, each
into I
segments
called segments of the chord. A secant segment is
a
that contains a chord of a circle, and has exactly one
outside the circle. An external segment is the part of
a
secant segment that is
the circte.
"til. SEGMENTS OF CHORDS THEOREM
ttre rnk"ior
i two chords intersect in
of a circle,
then the product of the lengths of the segments of one chord is equal
to the product of the lengths of the segments of the other chord. =
1)
Find x, ML and JK. t!
(x +sX
orn;+ \!, rl
.r(?+
r
^ \"" lv
x-5..-'t.J
4x
f,|l) = x(x+r) t-3 q
4xt?
=
lrl
CD
n L z j+3+3 +l
=[q
/-2+Sx
5/
EC. .tilPj
TK " 3+-s+3
=f)
SEGMENTS OF SECANTS THEOREM
lf two secant segments share the same endpoint
o*slA,c
a circle,
rhen the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment'
(Whole secant segment o outside segment) = (Whole secant segment o outside segment)
EA
.
EB =EC. eD
2l
Find the vplue of x.
F \ro
6(ro1 = 6o =
#:,-)
s(x+5)
Sx+a(
as :5*
Ff
Find the value of x.
b)
,y ,\
6x = strz)
6 ( r+) --
r L*tr)
Bt ='1* kqtq 35 =1{
6x = Lo
tffi-i
SEGMENTS OF SECANTS AND TANGENTS THEOREM
lf a secant segment and a tangent segment share an
ertlpoi
,.1-' U,JfS,I"
a circle,
hen the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.
(Tangent Segment)
2
E1==
o = (Whole secant segment outside segment)
4l Find the value of x.
a)*14a -
b)
ft4..................,#
X
I't'= 1 (r +x )
116= t{q f1il I$1 = -1:t
F\
x'= 4 ( {tl) xo= t ( rtl L d ssz
Ec '
ED
Geometry
10.7
- Write
and Graph Equations of Circles
Learning Target: By the end of today's lesson we will be able to successfully write equations of circles in the coordinate plane.
STANDARD EQUATION OF A CIRCLE The standard equation of a circle with
cenfc.r
(h, k)
and
rq,lius
r
is:
x-2.+52 "
(x-l-,\-* (,1-k\2 = rL
Yz.,\*^ {H'' c.e..k^ Coro) \l
-,F
EQUATION FOR RADIUS OF A CIRCLE
@=
(A;.lorr.r- +."^",U)
1) Write the equation of the circle shown. a)
+# equation:
2l
f = ?,
ro,J.r^s a
C"".lcr (ot o)
cu.+
t+ tt.*
Eouation:
Write the equation of the circle with the given information. a) With center (0, -5) and radius 3'7
xz+ (y+s)2
.,/(-3+l
z7
i-t U-=?b
b) With center (-3, -5) and radius 6.1 t+ [x+a) (I+s)z= ?-1.
tt
= l?.tq
3) The point (-3, 4) is on a circle with center (-t,2).
r:
)z+C+
Write the standard equation of the circle.
-e)'
-
(x+
; v@iTz;i"-
r)a (g-x)"=
=y'm
=ff :
A{5-
4l The point (-I,2J is on a circle with center (3, -3). Write the standard equation of the circle.
r: J(-r-a)1+(z+3)-
r= /ffi1o r2
{ffi
f=
',ltt
G
(x-
r)'+
(y+3)2
= '{ l
I
5)
Use the given equation to graph the circle. (x-2\'+ (y + 312 = 16
a)
"."tler
rq,lJqs
(a ,-a1
+
,---\ Cer,{
(-2, t)
rodirrS 3
Geometry
Name
LO.4
Hour
Date Find the value of the variables.
T
a)
2y+b4
=
b)
{.i
t-h;+ 6lt= l3 o
l8o
z*--llL
@
I
' ltfrl{"{-.'*-" v*\ 1,.^ j
ll4n"
\r'' \"^F l.!!l$o * = a.
\ '{\ # ""4 ["q' &s];r ;" t3y-11'-t+" ,'l \..) "IxU \* #
1x+63:\6o
5€ lgo
| +6
11 z ll-l
3y* 6+ = l?o
f;4
Find the value of x in each.
2l
10.7 Worksheet
F4l
3l-
4u-- tlb Y-- aq\
-
/\/-*\
3)
/\
4l
.--. x
/ L-*
l\
..^!
erL{F.s+r'
:5t
..".4. 1
.-r*- -.i3 ''{ rt2._7} -"\
''
Y
i
\"t
\*-.._.--
5+-11
e.
€t=x a
9{
=
3..
=E-
tso-+-l
7l
=q1"
aS+t( a : *
43
l2o+18
a
aS+t" 8L
s)
ln
l\#
*\ \_fi rl**
!13=ffi l.
5t(21
t\" j\{
t*
*
l
i
.@
U
ut artffil
Xt r** \.:. / \,, I --*
-
tor
(z) =l t"
fl
alo' 1
"FFI
-
{'-^""'qLL
,u*u\,/
lo,8-1 : *
a
1!:* l,
rtt\
H=x ). lI
=x
@
la.- rx t
tl
}. =x
:*
l
lJt- cr =X t
$ =y tx=16\
Match the equation of a circle with its description.
A.
fi**t*t
fi.
{:{:nlg'{{:i,
rlrcl]u*
i
ir
D.
{:rrrler {1, 51, r*rJius
"}
[6
E. **nl*r 1
"1" .51-
F.
{{}"
fi},
- l, +}- l'fidi*s E. **nl*t'{ -:" -i i" r*':trius "}
C te. ,l+,1=+ F
fA. n.2-,up,2*q
A .tu. i,x * ill + { r,- 4ll = ls B
fll+{r,+ t,.v * -i12 + {.r -
1.
E b
{"! +
es.
cr.
-
-1 |
_r
t]:+{r,-
-
{",t
'!I
-
-
..4
{':{:r}h.'n'
-tr
{
{"}1"
r'aJius 4
rrrq"lius
-1
{iive th* c*nt*r a,nd rsqliila *d t}re *ircle. zz. "a,2 + rJ ::s Ce,n*
o) r: 4 24. {,v-zI: *{...r,- 4:'t2=l"ti Ce^{cr (a,+1 r:{
ts.
{,,q
- i}: +
^r,1
- tfr Cc^Lr
(S,
Write the standard equation of the circle with the given center and radius.
irl
+4
{:'Lnt*r {t}.
e'}1.
ril({nrr: it+5t= 1
26.
{-enter {t*
{"r}.
rxdlurs
nn
{-'*"nr*r
28.
29.
CX- a)2+ } '(xt+ Yt= I
(-?r")
(b- q)t= {q : a {:i*ntrr {fi. il.ir nq,,Jius l{} Xt+ b = loO {il.
rrr^ r;rLliLrs
I
( x+
f,r,
...ir:
Use the giv,en in{-{rrffisti*n t& write th* *tandartl e,qu*ti-*rr gl. Tlr* *,enl,*,r is t{.t" {-t'l- ren** zr nr*illt *t1 lh* *,ir:r".|* is { -i, -{'l
th* sir*le.
32. Th* *r'1tr*e lx {.i, 4}, ;rnul I p;rl*[ ,.rtr ilt,-'r'tru'!u' rs ii" It tX - Z) r--| I J = ,/@:+ 6TA- =rfi-= Seterrnins thp diarr,*ter *f ths *lr*le with th* giu*tt eqetation' 34. {.r * f tl +-i,' +l_il =
3
z-=
.=rE
r=3
Graph each equation. SS. :i'i + l,f : -:i
SS.
{.,r
-
1
}:
*
t, l-=+
I
Jl.
,
( l1u)
l)l:4
';
Srtprrn,in* wheth,er th* trl*itrt li*s
{n-t}r+{y-S}s=25' 39. -1, ( ,r -21'+ (r- L)a = z5 N" za+ (-*)1' 15 4 { lL :zS }u 1"t r
-11
16
x"+ qt= xS
3:F=GTGI=ffi --s r?5
d=
sb{
a)'* )""
crn
the *irele ds*srlh*d by th* equation ilfi.
{,fa" t}
}
(c-r\'t ( 1-b)' : 2'5
Yu,s {2 + Ta* 2€ lLlQz25 ayzLS
(r'-t,
Geometry
Name 10.6
- 10.7 Review Worksheet
Show allwork for full credit!
Date
Fill in the blanks. Then find the value of x.
t. ... 3 =1.9
z. rl . 4 =i.
3x={151 3x--
E
4. +'
X :S.
*5
x
X
:+.
12
f,tl =3x
3x = t$B
@
8 =x
E
18
z
.---r..-l*
[= 3u
'Ee
,f--t;{to{*=go
i
-"-'
V-
nY"
..*,,i l-- X = lr.5l
\i*-./
Find the value of x.
5x = 6(ro)
6x= 6o
tr'la\
)*i ,lj
3(x+s)= tt(rl) Sxtg = ++
Jffi
zx=Bt5)
(x+q)-, ( I {,) \/s I |,.d;->*-5'
ri \t,*.-_-./'
tx+ lu = tl.B \1= ll).
Xz= 3 tz.t)
*t=
?r
1: 6{i Gl'^ -i"=
Stl'
i
15.
I-x-- 3
-
8= x (zx) 8*=
c{'r-
6t= ly231
-- x
z
t-ffiI
12
lll
t*=
\o.+
12". 1t1q
'
x(tt) \Lr(
@
|
Find the value of x in each
!4t
23x=zg(rs)
f,r\lJi t''.*
1s)
rlJ-r ,tl llx=G(*++l r'+
j ri
I l1 i|. 'tl
t
\
!.-
l;r'
\x=ts\
3(x+3): 2 (rz)
I
I
\;--*-1 .-,"/J
6
3*+q = z{
\*
€
(to)
6 (x+5) =
\
jh/
il
6l'+3o= 6o
p
t*=
rl
3o
t5(3:t+rS) = 2o ( x+zo1 t5rt + 2zS . ?.a* +*o o
= 5
25x = 115
5t
\
.{-",.*i
*_,,
2
{=
2
lf=
9 (tt")
-152 5:t
F
9 ( a*+r)
z 3[rl 1q* : ?!x
x= 1t+
loo? Ex .t aS
/
Llt
5(tz)
"2.
r1-... r,. A'+ h .j'6 /
ztl
rf. s (xts)
231
i
EI
(ro ?
**JF**t "\;, \ i1r'
I
t2*= b* t3"q Gx--
@
$h-r\i
1\
k+s) 6oo 5 *+ t5
3*=tS 20)
/\
16)
?LS
Xt
ql [-x= l--r
241
\^
\\#T;-\-./ iG lYr,F\
r
oo"if
orr.i
I
LiA" = ?t(z*+to)
1r3" Ltf, +zo* 5f-aox=ts
5(*-*'=
h
26) Write the standard equation of each circle below.
a)
Center: (0, O), radius = 9
b)
Center: (2,7), diameter = 4
(x-z) d)
Center: (-1, 5), radius = 3
(
i)"*
(3- 5)"= 1
e)
i
(t-r)-r'=
[
Center:(-8, 11), Point (-5, 6)
6r+s\2+ (tt-s)a
{-4'*_5"
c)
n( xrt)iCg - ni! rt
Center: (-8, 1L), d = 9
(x *S)L1 (..t -rl)'= ? I
f)
Center: (0,0), point (3,4)
X1rq"=
zi
.ITFTL
G 5
Write the standard equation of each circle below. 271
281
i: :"'-'*t' l
j'-
|
1
li
i
L
(x-a)tr (q -t)"=
f* q'= 36
.t I--- i- :l
I
)1
'r
a
Use the equation of a circle to find the center, radius and diameter.
19)
- 5)'+ (Y -
(x
30)
6)2 = l6Q
C= (-1,1')
C=(qra) l-'r
r3l
IO
J= 1
Ir. $o *raph the *qtrwti*ar" 3"] .r,"r +..r.2 * ]5
]sl
"i
,f
4^
*
+.l'l = ,{
tr12
C-- (t,o)
c=
V=L
!!
i:
14I
1.x
(o,O Y=5
l
I
31) x'+(y-4l,'=8I
(x+9)2 +(y-2\2=1
4v'& }.+l
*
ls]
q3
t"';
"* 3r)'& r,'; +
1!3
c'- (o,^z') 1\
/j
,
I
3
t",
-17
{,
C- 36! \\ rite rhc gfanr"lal'ad ri il,
r--
I
*qrartii*rn erf x u*irc$*
lvilh its
Lr;ind t'ndiu*,1
t' *.E ' = fr .I !-
\
i'1. t] * l:l = j
h. --+ , f} fr =t 3T] Hiritr: flru *insia{srr{ e:qrrutiein r:l'u tirr'l+ lvitfu
ur-,tl{t-:r
f-}.
*,Si *nd rrtclirsi h
*}.r: * r..v*ai: =* ?r. tr -;11: * f .1,* 4l: = 6
u.
L
{.r
3S] Tlrr: g-fi*l'rilir.nl ilqettlilrat +f * *ir"-l* with
;r. h.
a-.t
{.r
*.
4il *
i.S + -i
i:
=
tsrlt*r t-'i- .il iur,"l t;rq!1gf
I
7
-4}J * f "11;ii1 =-{*
tt. ll+4l+{r-"3}ai
*'1
(:
( a,- r)
(=
L
11
