answers (lesson 11-1) · rationalizing the denominator ... pre-activity how are radical expressions...
TRANSCRIPT
© Glencoe/McGraw-Hill A2 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Sim
plify
ing
Rad
ical
Exp
ress
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-1
11-1
©G
lenc
oe/M
cGra
w-H
ill64
3G
lenc
oe A
lgeb
ra 1
Lesson 11-1
Prod
uct
Prop
erty
of
Squa
re R
oots
The
Pro
du
ct P
rop
erty
of
Squ
are
Roo
tsan
dpr
ime
fact
oriz
atio
n ca
n be
use
d to
sim
plif
y ex
pres
sion
s in
volv
ing
irra
tion
al s
quar
e ro
ots.
Whe
n yo
u si
mpl
ify
radi
cal e
xpre
ssio
ns w
ith
vari
able
s,us
e ab
solu
te v
alue
to
ensu
reno
nneg
ativ
e re
sult
s.
Pro
duct
Pro
pert
y of
Squ
are
Roo
tsF
or a
ny n
umbe
rs a
and
b, w
here
a!
0 an
d b
!0,
!ab"
"!
a"#
!b".
Sim
pli
fy !
180
".
!18
0"
"!
2 #
2 #
"3
#3
"#
5"P
rime
fact
oriz
atio
n of
180
"!
22 "#
!32 "
#!
5"P
rodu
ct P
rope
rty
of S
quar
e R
oots
"2
#3
#!
5"S
impl
ify.
"6!
5"S
impl
ify.
Sim
pli
fy !
120a
2"
!b5
!"
c4 ".
!12
0a2
"#
b5#
"c4 "
"!
23#
3"
#5
#a
"2
#b5
"#
c4 ""
!22 "
#!
2"#
!3"
#!
5"#
!a2 "
#!
b4#
b"
#!
c4 ""
2 #
!2"
#!
3"#
!5"
#|a
|#b2
#!
b"#
c2
"2
|a|b
2 c2 !
30b
"
Sim
pli
fy.
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3.!
60"4.
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162
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2"
9.!
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4"
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300a
4"
12.!
128c
6"
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2"
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2 | y|
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| !6c"
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72a6
b"
3 c2
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x2y
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8"
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6 z2
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3 bc|
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5y"7x
2 | y3 z
| !2"
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill64
4G
lenc
oe A
lgeb
ra 1
Quo
tien
t Pr
oper
ty o
f Sq
uare
Roo
tsA
fra
ctio
n co
ntai
ning
rad
ical
s is
in s
impl
est
form
if n
o ra
dica
ls a
re le
ft in
the
den
omin
ator
.The
Qu
otie
nt
Pro
per
ty o
f S
quar
e R
oots
and
rati
onal
izin
g th
e d
enom
inat
orca
n be
use
d to
sim
plif
y ra
dica
l exp
ress
ions
tha
tin
volv
e di
visi
on.W
hen
you
rati
onal
ize
the
deno
min
ator
,you
mul
tipl
y th
e nu
mer
ator
and
deno
min
ator
by
a ra
dica
l exp
ress
ion
that
giv
es a
rat
iona
l num
ber
in t
he d
enom
inat
or.
Quo
tient
Pro
pert
y of
Squ
are
Roo
tsF
or a
ny n
umbe
rs a
and
b, w
here
a!
0 an
d b
!0,
#$"
.
Sim
pli
fy #$
.
#$"#$
"S
impl
ify th
e nu
mer
ator
and
den
omin
ator
.
"#
Mul
tiply
by
to r
atio
naliz
e th
e de
nom
inat
or.
"P
rodu
ct P
rope
rty
of S
quar
e R
oots
Sim
pli
fy.
1.2.
3.4.
5
5.2
6.#$
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2 #
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$ 3 #
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4 #
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#5
56 $ 45
56 # 45
!a"
$ !b"
a $ b
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sim
plify
ing
Rad
ical
Exp
ress
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-1
11-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 11-1)
© Glencoe/McGraw-Hill A3 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Sim
plify
ing
Rad
ical
Exp
ress
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-1
11-1
©G
lenc
oe/M
cGra
w-H
ill64
5G
lenc
oe A
lgeb
ra 1
Lesson 11-1
Sim
pli
fy.
1.!
28"2!
7"2.
!40"
2!10"
3.!
72"6!
2"4.
!99"
3!11"
5.!
2"#
!10"
2!5"
6.!
5"#
!60"
10!
3"
7.3!
5"#
!5"
158.
!6"
#4!
24"48
9.2!
3"#
3!15"
18!
5"10
.!16
b4"
4b2
11.!
81c2
d"
4 "9|
c| d
212
.!40
x4y
"6 "
2x2 | y
3 | !10"
13.!
75m
5 n"
2 "5m
2 | n| !
3m"14
. #$
15. #
$16
. #$#
#$
17. #
$18
. #$
19. #
$20
. #$
21.
22.
6 $
3!3"
23.
24.
12 $
4!2"
## 7
4$ 3
&!
2"35
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7"#
# 425
$ 7 %
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3$ 2
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5"#
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2!3"
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4h $ 5!
3q"#
6q $ 12
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#7
1 $ 36 $ 7
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1 $ 6
!15"
#3
5 $ 3
©G
lenc
oe/M
cGra
w-H
ill64
6G
lenc
oe A
lgeb
ra 1
Sim
pli
fy.
1.!
24"2!
6"2.
!60"
2!15"
3.!
108
"6!
3"4.
!8"
#!
6"4!
3"
5.!
7"#
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6.3!
12"#
5!6"
90!
2"
7.4!
3"#
3!18"
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6"8.
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su3
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9.!
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6"
y4z5
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3 | y2 z
2 !3z"
11.!
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| n2 o
2 !14
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12.
13. #
$14
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15. #
$##$
16. #
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17.
18. #
$19
. #$
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21.
22.
23.
24.
25. S
KY
DIV
ING
Whe
n a
skyd
iver
jum
ps fr
om a
n ai
rpla
ne,t
he t
ime
tit
take
s to
free
fall
a
give
n di
stan
ce c
an b
e es
tim
ated
by
the
form
ula
t"#$
,whe
re t
is in
sec
onds
and
sis
in
met
ers.
If J
ulie
jum
ps fr
om a
n ai
rpla
ne,h
ow lo
ng w
ill it
tak
e he
r to
free
fall
750
met
ers?
abou
t 12.
4 s
MET
EOR
OLO
GY
For
Exe
rcis
es 2
6 an
d 2
7,u
se t
he
foll
owin
g in
form
atio
n.
To e
stim
ate
how
long
a t
hund
erst
orm
will
last
,met
eoro
logi
sts
can
use
the
form
ula
t"#$
,whe
re t
is t
he t
ime
in h
ours
and
dis
the
dia
met
er o
f th
e st
orm
in m
iles.
26.A
thu
nder
stor
m is
8 m
iles
in d
iam
eter
.Est
imat
e ho
w lo
ng t
he s
torm
will
last
.Giv
e yo
uran
swer
in s
impl
ifie
d fo
rm a
nd a
s a
deci
mal
.#8 !
93 " #
h %
1.5
h27
.Will
a t
hund
erst
orm
tw
ice
this
dia
met
er la
st t
wic
e as
long
? E
xpla
in.
No;
it w
ill la
st a
bout
4.4
h,o
r ne
arly
3 t
imes
as
long
.
d3$ 21
6
2s $ 9.8
3!7"
" 9
!21"
## 26
3!7"
$$
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27"5!
7""
5!3"
## 4
5$
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4!3"
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9ab
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y|
4y $ 3y2
3!2x"
#x
218 $ x3
!6k"
#4
!3k"
$ !8"
!11"
#11
7 $ 111 $ 7
!15"
#5
4 $ 53 $ 4
!10"
#8
5 $ 32!
5"#
52 $ 10
2!3"
#3
!8"
$ !6"
Pra
ctic
e (A
vera
ge)
Sim
plify
ing
Rad
ical
Exp
ress
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-1
11-1
Answers (Lesson 11-1)
© Glencoe/McGraw-Hill A4 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csS
impl
ifyin
g R
adic
al E
xpre
ssio
ns
NA
ME
____
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____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-1
11-1
©G
lenc
oe/M
cGra
w-H
ill64
7G
lenc
oe A
lgeb
ra 1
Lesson 11-1
Pre-
Act
ivit
yH
ow a
re r
adic
al e
xpre
ssio
ns
use
d i
n s
pac
e ex
plo
rati
on?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-1 a
t th
e to
p of
pag
e 58
6 in
you
r te
xtbo
ok.
Supp
ose
you
wan
t to
cal
cula
te t
he e
scap
e ve
loci
ty f
or a
spa
cecr
aft
taki
ng o
fffr
om t
he p
lane
t M
ars.
Whe
n yo
u su
bsti
tute
num
bers
in t
he f
orm
ula,
whi
chnu
mbe
r is
sur
e to
be
the
sam
e as
in t
he c
alcu
lati
on f
or t
he e
scap
e ve
loci
tyfo
r a
spac
ecra
ft t
akin
g of
f fr
om E
arth
?th
e va
lue
of G
Read
ing
the
Less
on
1.a.
How
can
you
tel
l tha
t th
e ra
dica
l exp
ress
ion
!28
x2y
"4 "i
s no
t in
sim
ples
t fo
rm?
The
radi
cand
con
tain
s pe
rfec
t sq
uare
fac
tors
oth
er t
han
1.
b.To
sim
plif
y !
28x2
y"
4 ",yo
u fi
rst
find
the
of
28x
2 y4 .
You
then
app
ly t
he
.In
this
cas
e,
!4
#7
#"
x2#
y"
4 "is
equa
l to
the
prod
uct
.You
can
sim
plif
y ag
ain
to g
et a
fin
al a
nsw
er o
f 2 |
x|y2
!7".
2.W
hy is
it c
orre
ct t
o w
rite
!y4 "
"y2
,wit
h no
abs
olut
e va
lue
sign
,but
not
cor
rect
to
wri
te
!x2 "
"x?
Sam
ple
answ
er:T
he s
quar
e of
y2
is y
4an
d th
e ex
pres
sion
s !
y4 "
and
y2
both
rep
rese
nt p
ositi
ve n
umbe
rs fo
r al
l val
ues
of y
.Alth
ough
it is
tru
e th
at t
he s
quar
e of
xis
x2 ,
whe
n x
is le
ss t
han
0,!
x2 "re
pres
ents
a
posi
tive
quan
tity
and
xre
pres
ents
a n
egat
ive
quan
tity.
3.W
hat
met
hod
wou
ld y
ou u
se t
o si
mpl
ify
?
ratio
naliz
ing
the
deno
min
ator
4.W
hat
shou
ld y
ou d
o to
wri
te t
he c
onju
gate
of
a bi
nom
ial o
f th
e fo
rm a
!b"
%c!
d"? T
o w
rite
the
con
juga
te o
f a
bino
mia
l of
the
form
a!
b"&
c!d"?
Cha
nge
the
plus
sig
n to
a m
inus
sig
n;ch
ange
the
min
us s
ign
to a
pl
us s
ign.
Hel
ping
You
Rem
embe
r
5.W
hat
shou
ld y
ou r
emem
ber
to c
heck
for
whe
n yo
u w
ant
to d
eter
min
e if
a r
adic
alex
pres
sion
is in
sim
ples
t fo
rm?
Sam
ple
answ
er:C
heck
rad
ican
ds fo
r pe
rfec
t sq
uare
s an
d fr
actio
ns,a
ndch
eck
frac
tions
for
radi
cals
in t
he d
enom
inat
or.
!12
t"
# !15"!
4"!
!7"
!!
x2 "
!!
y4 "
Pro
duct
Pro
pert
y of
Squ
are
Roo
ts
prim
e fa
ctor
izat
ion
©G
lenc
oe/M
cGra
w-H
ill64
8G
lenc
oe A
lgeb
ra 1
Squ
ares
and
Squ
are
Roo
ts F
rom
a G
raph
The
gra
ph o
f y"
x2ca
n be
use
d to
fin
d th
e sq
uare
s an
d sq
uare
ro
ots
of n
umbe
rs.
To f
ind
the
squa
re o
f 3,
loca
te 3
on
the
x-ax
is.T
hen
find
its
corr
espo
ndin
g va
lue
on t
he y
-axi
s.
The
arr
ows
show
tha
t 32
"9.
To f
ind
the
squa
re r
oot
of 4
,fir
st lo
cate
4 o
n th
e y-
axis
.The
n fi
nd
its
corr
espo
ndin
g va
lue
on t
he x
-axi
s.Fo
llow
ing
the
arro
ws
on t
he
grap
h,yo
u ca
n se
e th
at !
4""
2.
A s
mal
l par
t of
the
gra
ph a
t y
"x2
is s
how
n be
low
.A 1
:10
rati
o fo
r un
it le
ngth
on
the
y-ax
is t
o un
it le
ngth
on
the
x-ax
is is
use
d.
Fin
d !
11".
The
arr
ows
show
tha
t !
11""
3.3
to t
he n
eare
st t
enth
.
Use
th
e gr
aph
abo
ve t
o fi
nd
eac
h o
f th
e fo
llow
ing
to t
he
nea
rest
wh
ole
nu
mbe
r.
1.1.
522
2.2.
727
3.0.
921
4.3.
6213
5.4.
2218
6.3.
9215
Use
th
e gr
aph
abo
ve t
o fi
nd
eac
h o
f th
e fo
llow
ing
to t
he
nea
rest
ten
th.
7.!
15"3.
98.
!8"
2.8
9.!
3"1.
7
10.!
5"2.
211
.!14"
3.7
12.!
17"4.
1
12
11 1020
34
3.3
Ox
y
x
y O
En
rich
men
t
NA
ME
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__D
ATE
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RIO
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11-1
11-1
Exam
ple
Exam
ple
Answers (Lesson 11-1)
© Glencoe/McGraw-Hill A5 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Ope
ratio
ns w
ith R
adic
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-2
11-2
©G
lenc
oe/M
cGra
w-H
ill64
9G
lenc
oe A
lgeb
ra 1
Lesson 11-2
Add
and
Sub
trac
t Ra
dica
l Exp
ress
ions
Whe
n ad
ding
or
subt
ract
ing
radi
cal
expr
essi
ons,
use
the
Ass
ocia
tive
and
Dis
trib
utiv
e P
rope
rtie
s to
sim
plif
y th
e ex
pres
sion
s.If
radi
cal e
xpre
ssio
ns a
re n
ot in
sim
ples
t fo
rm,s
impl
ify
them
.
Sim
pli
fy 1
0 !6"
!5!
3""
6!3"
!4!
6".10
!6"
!5!
3""
6!3"
!4!
6"#
(10
!4)
!6"
"(!
5 "
6)!
3"A
ssoc
iativ
e an
d D
istr
ibut
ive
Pro
pert
ies
#6!
6""
!3"
Sim
plify
.
Sim
pli
fy 3
!12"
"5!
75".
3!12"
"5!
75"#
3!22
$3
""
5!52
$3
"S
impl
ify.
#3
$2!
3""
5 $
5!3"
Sim
plify
.
#6!
3""
25!
3"S
impl
ify.
#31
!3"
Dis
trib
utiv
e P
rope
rty
Sim
pli
fy e
ach
exp
ress
ion
.
1.2!
5""
4!5"
6!5"
2.!
6"!
4!6"
!3!
6"
3.!
8"!
!2"
!2"
4.3!
75""
2!5"
15!
3""
2!5"
5.!
20""
2!5"
!3!
5"!
5"6.
2!3"
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6"!
5!3"
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3""
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7.!
12""
2!3"
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3"!
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6""
3!2"
!!
50""
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2"
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8a"!
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6"
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180
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4!45"
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125
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2 #$"
#$"
18. #
$"3!
3"!
4 #$
"28
!3 "
#12
!6 "
#3
1 % 122 % 3
!3 "
#3
23!
5 "#
51 % 3
1 % 5
8 !3 "
#3
1 % 3
17!
6"#
61 % 6
7 !3 "
#3
1 % 34!
3"#
31 % 3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill65
0G
lenc
oe A
lgeb
ra 1
Mul
tipl
y Ra
dica
l Exp
ress
ions
Mul
tipl
ying
tw
o ra
dica
l exp
ress
ions
wit
h di
ffer
ent
radi
cand
s is
sim
ilar
to m
ulti
plyi
ng b
inom
ials
.
Mu
ltip
ly (3
!2"
!2!
5")(4
!20"
"!
8").
Use
the
FO
IL m
etho
d.(3
!2"
!2!
5")(4
!20"
"!
8")#
(3!
2")(4
!20"
)"(3
!2")
(!8")
"(!
2!5")
(4!
20")"
(!2!
5")(!
8")#
12!
40""
3!16"
!8!
100
"!
2!40"
Mul
tiply
.
#12
!22
$10
""
3 $
4 !
8 $
10 !
2!22
$10
"S
impl
ify.
#24
!10"
"12
!80
!4!
10"S
impl
ify.
#20
!10"
!68
Com
bine
like
term
s.
Fin
d e
ach
pro
du
ct.
1.2(
!3"
"4!
5")2.
!6"(
!3"
!2!
6")2!
3""
8!5"
3!2"
!12
3.!
5"(!
5"!
!2")
4.!
2"(3!
7""
2!5")
5 !
!10"
3!14"
"2!
10"
5.(2
!4!
2")(2
"4!
2")6.
(3 "
!6")
2
!28
15 "
6!6"
7.(2
!2!
5")2
8.3!
2"(!
8""
!24"
)24
!8!
5"12
"12
!3"
9.!
8"(!
2""
5!8")
10.(
!5"
!3!
2")(!
5""
3!2")
44!
13
11.(!
3""
!6")
212
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2"!
2!3")
2
9 "
6!2"
14 !
4!6"
13.(!
5"!
!2")
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6")14
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8"!
!2")
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6")!
10"!
2 "
!30"
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3"!
6""
2!3"
15.(!
5"!
!18"
)(7!
5""
!3")
16.(
2!3"
!!
45")(!
12""
2!6")
35 "
!15"
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10"!
3!6"
12 !
6!15"
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!2"
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30"
17.(2
!5"
!2!
3")(!
10""
!6")
18.(
!2"
"3!
3")(!
12"!
4!8")
4!2"
2 !
22!
6"
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Ope
ratio
ns w
ith R
adic
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-2
11-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 11-2)
© Glencoe/McGraw-Hill A6 Glencoe Algebra 1
Skil
ls P
ract
ice
Ope
ratio
ns w
ith R
adic
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-2
11-2
©G
lenc
oe/M
cGra
w-H
ill65
1G
lenc
oe A
lgeb
ra 1
Lesson 11-2
Sim
pli
fy e
ach
exp
ress
ion
.
1.7!
7"&
2!7"
5!7"
2.3!
13"%
7!13"
10!
13"
3.6!
5"&
2!5"
%8!
5"12
!5"
4.!
15"%
8!15"
&12
!15"
"3!
15"
5.12
!c"
&9!
c"3!
c"6.
9!6a"
&11
!6a"
%4!
6a"2!
6a"
7.!
44"&
!11"
!11"
8.!
28"%
!63"
5!7"
9.4!
3"%
2!12"
8!3"
10.8
!54"
&4!
6"20
!6"
11.!
27"%
!48"
%!
12"9!
3"12
.!72"
%!
50"&
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9!2"
13.!
180
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5!5"
%!
20"3!
5"14
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24"%
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96"36
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20"&
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16.2
!13"
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2"&
5!13"
%!
2"8!
2"$
4!5"
"3!
13"$
5!2"
Fin
d e
ach
pro
du
ct.
17.!
2"(!
8"%
!6")
4 $
2!3"
18.!
5"(!
10"&
!3")
5!2"
"!
15"
19.!
6"(3!
2"&
2!3")
6!3"
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2"20
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)18!
2"$
12!
30"
21.(4
%!
3")(4
&!
3")13
22.(
2 &
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210
"4!
6"
23.(!
8"%
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(!5"
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3")24
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6"%
4!5")
(4!
3"&
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10"$
3!6"
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2"$
14!
15"
©G
lenc
oe/M
cGra
w-H
ill65
2G
lenc
oe A
lgeb
ra 1
Sim
pli
fy e
ach
exp
ress
ion
.
1.8!
30"&
4!30"
4!30"
2.2!
5"%
7!5"
&5!
5"4!
5"
3.7!
13x
"&
14!
13x
"%
2!13
x"
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13x
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2!45"
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20"14
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40"&
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90"4!
10"6.
2!32"
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2"
7.!
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300
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2"$
13!
3"8.
5!8"
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20"&
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6!2"
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9.!
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11.5
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63"12
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11!
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Fin
d e
ach
pro
du
ct.
13. !
6"(!
10"%
!15"
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15"$
3!10"
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5"(5!
2"&
4!8")
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3!12"
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8")12
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$20
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5 &
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!15"
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!8"
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12")(!
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)36
$14
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19.(!
2"%
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4!3"
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5")(3
!10"
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!3"
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10"2!
30"$
30!
2"
SOU
ND
For
Exe
rcis
es 2
1 an
d 2
2,u
se t
he
foll
owin
g in
form
atio
n.
The
spe
ed o
f sou
nd V
in m
eter
s pe
r se
cond
nea
r E
arth
’s su
rfac
e is
giv
en b
y V
"20
!t
%27
"3",
whe
re t
is t
he s
urfa
ce t
empe
ratu
re in
deg
rees
Cel
sius
.
21.W
hat
is t
he s
peed
of
soun
d ne
ar E
arth
’s s
urfa
ce a
t 15
°C a
nd a
t 2°
C in
sim
ples
t fo
rm?
240!
2"m
/s,1
00!
11"m
/s
22.H
ow m
uch
fast
er is
the
spe
ed o
f so
und
at 1
5°C
tha
n at
2°C
?24
0!2"
"10
0!11"
%7.
75 m
/s
GEO
MET
RYF
or E
xerc
ises
23
and
24,
use
th
e fo
llow
ing
info
rmat
ion
.A
rec
tang
le is
5!
7"%
2!3"
cent
imet
ers
long
and
6!
7"&
3!3"
cent
imet
ers
wid
e.
23.F
ind
the
peri
met
er o
f th
e re
ctan
gle
in s
impl
est
form
.22
!7"
"2!
3"cm
24.F
ind
the
area
of t
he r
ecta
ngle
in s
impl
est
form
.19
2 "
3!21"
cm217
!2"
#2
1 $ 26!
14"#
72 $ 7
Pra
ctic
e (A
vera
ge)
Ope
ratio
ns w
ith R
adic
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-2
11-2
Answers (Lesson 11-2)
© Glencoe/McGraw-Hill A7 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csO
pera
tions
with
Rad
ical
Exp
ress
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-2
11-2
©G
lenc
oe/M
cGra
w-H
ill65
3G
lenc
oe A
lgeb
ra 1
Lesson 11-2
Pre-
Act
ivit
yH
ow c
an y
ou u
se r
adic
al e
xpre
ssio
ns
to d
eter
min
e h
ow f
ar a
per
son
can
see
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-2 a
t th
e to
p of
pag
e 59
3 in
you
r te
xtbo
ok.
Supp
ose
you
subs
titu
te t
he h
eigh
ts o
f the
Sea
rs T
ower
and
the
Em
pire
Sta
teB
uild
ing
into
the
form
ula
to fi
nd h
ow fa
r yo
u ca
n se
e fr
om a
top
each
bui
ldin
g.W
hat
oper
atio
n sh
ould
you
the
n us
e to
det
erm
ine
how
muc
h fa
rthe
r yo
u ca
nse
e fr
om t
he S
ears
Tow
er t
han
from
the
Em
pire
Sta
te B
uild
ing?
subt
ract
ion
Read
ing
the
Less
on
1.In
dica
te w
heth
er t
he f
ollo
win
g ex
pres
sion
s ar
e in
sim
ples
t fo
rm.E
xpla
in y
our
answ
er.
a.6!
3"!
!12"
No;
12 c
an b
e si
mpl
ified
to
!22 "
!!
3"or
2!
3".
b.12
!6"
"7!
10"
Yes;
both
the
add
ends
are
rad
ical
exp
ress
ions
in s
impl
est
form
,the
radi
cand
s ar
e di
ffer
ent,
and
ther
e ar
e no
com
mon
fac
tors
.
2.B
elow
the
wor
ds F
irst
ter
ms,
Out
er t
erm
s,In
ner
term
s,an
d L
ast
term
s,w
rite
the
pr
oduc
ts y
ou w
ould
use
to
sim
plif
y th
e ex
pres
sion
(2!
15""
3!15"
)(6!
3"!
5!2")
.
Fir
st t
erm
sO
uter
ter
ms
Inne
r te
rms
Las
t te
rms
""
"
Hel
ping
You
Rem
embe
r
3.H
ow c
an y
ou u
se w
hat
you
know
abo
ut a
ddin
g an
d su
btra
ctin
g m
onom
ials
to
help
you
rem
embe
r ho
w t
o ad
d an
d su
btra
ct r
adic
al e
xpre
ssio
ns?
Sam
ple
answ
er:C
heck
tha
t th
e ad
dend
s ha
ve b
een
sim
plifi
ed.N
ext,
grou
p ad
dend
s th
at in
volv
e lik
e ra
dica
ls.T
hen
use
the
Dis
trib
utiv
eP
rope
rty
to c
ombi
ne t
he a
dden
ds t
hat
invo
lve
like
radi
cals
.
( 3!
5")( "
5!2")
( 3!
5")( 6
!3")
( 2!
15")( "
5!2")
( 2!
15")( 6
!3")
©G
lenc
oe/M
cGra
w-H
ill65
4G
lenc
oe A
lgeb
ra 1
The
Whe
el o
f Th
eodo
rus
The
Gre
ek m
athe
mat
icia
ns w
ere
intr
igue
d by
pro
blem
s of
re
pres
enti
ng d
iffe
rent
num
bers
and
exp
ress
ions
usi
ng
geom
etri
c co
nstr
ucti
ons.
The
odor
us,a
Gre
ek p
hilo
soph
er w
ho li
ved
abou
t 42
5 B.C
.,is
sa
id t
o ha
ve d
isco
vere
d a
way
to
cons
truc
t th
e se
quen
ce
!1",
!2",
!3",
!4",
….
The
beg
inni
ng o
f hi
s co
nstr
ucti
on is
sho
wn.
You
star
t w
ith
anis
osce
les
righ
t tr
iang
le w
ith
side
s 1
unit
long
.
Use
th
e fi
gure
abo
ve.W
rite
eac
h l
engt
h a
s a
rad
ical
exp
ress
ion
in
sim
ple
st f
orm
.
1.lin
e se
gmen
t A
O!
1"2.
line
segm
ent
BO
!2"
3.lin
e se
gmen
t C
O!
3"4.
line
segm
ent
DO
!4"
5.D
escr
ibe
how
eac
h ne
w t
rian
gle
is a
dded
to
the
figu
re.
Dra
w a
new
sid
e of
leng
th 1
at r
ight
ang
les
to t
he la
st h
ypot
enus
e. T
hen
draw
the
new
hyp
oten
use.
6.T
he le
ngth
of
the
hypo
tenu
se o
f th
e fi
rst
tria
ngle
is !
2".Fo
r th
e se
cond
tri
angl
e,th
ele
ngth
is !
3".W
rite
an
expr
essi
on fo
r th
e le
ngth
of t
he h
ypot
enus
e of
the
nth
tri
angl
e.
!n
#1
"7.
Show
tha
t th
e m
etho
d of
con
stru
ctio
n w
ill a
lway
s pr
oduc
e th
e ne
xt n
umbe
r in
the
sequ
ence
.(H
int:
Fin
d an
exp
ress
ion
for
the
hypo
tenu
se o
f th
e (n
"1)
th t
rian
gle.
)
!(!
n")2
"#
(1)
"2 "o
r !
n#
1"
8.In
the
spa
ce b
elow
,con
stru
ct a
Whe
el o
f The
odor
us.S
tart
wit
h a
line
segm
ent
1ce
ntim
eter
long
.Whe
n do
es t
he W
heel
sta
rt t
o ov
erla
p?
afte
r le
ngth
!18"
1
1
1
1 OAB
CD
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-2
11-2
Answers (Lesson 11-2)
© Glencoe/McGraw-Hill A8 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Rad
ical
Equ
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-3
11-3
©G
lenc
oe/M
cGra
w-H
ill65
5G
lenc
oe A
lgeb
ra 1
Lesson 11-3
Radi
cal E
quat
ions
Equ
atio
ns c
onta
inin
g ra
dica
ls w
ith
vari
able
s in
the
rad
ican
d ar
eca
lled
rad
ical
equ
atio
ns.
The
se c
an b
e so
lved
by
firs
t us
ing
the
follo
win
g st
eps.
Ste
p 1
Isol
ate
the
radi
cal o
n on
e si
de o
f the
equ
atio
n.S
tep
2S
quar
e ea
ch s
ide
of th
e eq
uatio
n to
elim
inat
e th
e ra
dica
l.
Sol
ve 1
6 %
for
x.
16 "
Orig
inal
equ
atio
n
2(16
) "
2 %&
Mul
tiply
eac
h si
de b
y 2.
32 "
!x"
Sim
plify
.
(32)
2"
(!x")
2S
quar
e ea
ch s
ide.
1024
"x
Sim
plify
.
The
sol
utio
n is
102
4,w
hich
che
cks
in t
heor
igin
al e
quat
ion.
!x"
$2
!x"
$2
!x"
#2
Sol
ve !
4x"
"7"
$2
%7.
!4x
&7
"%
2 "
7O
rigin
al e
quat
ion
!4x
&7
"%
2 &
2 "
7 &
2S
ubtr
act 2
from
eac
h si
de.
!4x
&7
""
5S
impl
ify.
(!4x
&7
")2
"52
Squ
are
each
sid
e.
4x&
7 "
25S
impl
ify.
4x&
7 %
7 "
25 %
7A
dd 7
to e
ach
side
.
4x"
32S
impl
ify.
x"
8D
ivid
e ea
ch s
ide
by 4
.
The
sol
utio
n is
8,w
hich
che
cks
in t
he o
rigi
nal
equa
tion
.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
.
1.!
a""
864
2.!
a"%
6 "
3267
63.
2!x"
"8
16
4.7
"!
26 &
"n"
"23
5.!
&a
""
6"
366.
!3r
2"
"3
&!
3"
7.2!
3""
!y"
128.
2!3a"
&2
"7
69.
!x
&4
""
640
10.!
2c%
3"
"5
1111
.!3b
&"
2"%
19 "
249
12.!
4x&
1"
"3
13.!
3r%
2"
"2!
3"14
. #$"
15. #
$"4
128
16.!
6x2
%"
5x""
2,"
17. #
$%6
"8
1218
.2#$
%3
"11
262 # 3
3x $ 5x $ 3
4 # 31 # 2
x $ 81 # 2
1 $ 2x $ 2
10 # 3
5 # 2
3 # 4
©G
lenc
oe/M
cGra
w-H
ill65
6G
lenc
oe A
lgeb
ra 1
Extr
aneo
us S
olut
ions
To s
olve
a r
adic
al e
quat
ion
wit
h a
vari
able
on
both
sid
es,y
oune
ed t
o sq
uare
eac
h si
de o
f th
e eq
uati
on.S
quar
ing
each
sid
e of
an
equa
tion
som
etim
espr
oduc
es e
xtra
neo
us
solu
tion
s,or
sol
utio
ns t
hat
are
not
solu
tion
s of
the
ori
gina
l equ
atio
n.T
here
fore
,it
is v
ery
impo
rtan
t th
at y
ou c
heck
eac
h so
luti
on.
Sol
ve !
x$
3"
%x
"3.
!x
%3
""
x&
3O
rigin
al e
quat
ion
(!x
%3
")2
"(x
&3)
2S
quar
e ea
ch s
ide.
x%
3 "
x2&
6x%
9S
impl
ify.
0 "
x2&
7x%
6S
ubtr
act x
and
3 fr
om e
ach
side
.
0 "
(x&
1)(x
&6)
Fac
tor.
x&
1 "
0or
x&
6 "
0Z
ero
Pro
duct
Pro
pert
y
x"
1x
"6
Sol
ve.
CH
ECK
!x
%3
""
x&
3!
x%
3"
"x
&3
!1
%3
"!
1 &
3!
6 %
3"
!6
&3
!4"
!&
2!
9"!
32
'&
23
"3
✓
Sinc
e x
"1
does
not
sat
isfy
the
ori
gina
l equ
atio
n,x
"6
is t
he o
nly
solu
tion
.
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
.
1.!
a""
a0,
12.
!a
%6
""
a3
3.2!
x""
x0,
4
4.n
"!
2 &
n"
15.
!&
a"
"a
06.
!10
&"
6k"%
3 "
k'
7.!
y&
1"
"y
&1
1,2
8.!
3a&
"2"
"a
1,2
9.!
x%
2"
"x
2
10.!
2c%
5"
"c
&5
1011
.!3b
%"
6""
b%
21
12.!
4x&
4"
"x
2
13.r
%!
2 &
r"
"2
1,2
14.!
x2%
1"
0x""
x%
48
15.&
2 #$"
15'
16.!
6x2
&"
4x""
x%
217
.!2y
2&
"64"
"y
18.!
3x2
%"
12x
%"
1""
x%
5
",2
8"
4,3
2 # 5
x $ 8
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Rad
ical
Equ
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-3
11-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 11-3)
© Glencoe/McGraw-Hill A9 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Rad
ical
Equ
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-3
11-3
©G
lenc
oe/M
cGra
w-H
ill65
7G
lenc
oe A
lgeb
ra 1
Lesson 11-3
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
.
1.!
f""
749
2.!
&x
""
5"
25
3.!
5p""
1020
4.!
4y""
69
5.2!
2""
!u"
86.
3!5"
"!
&n
""
45
7.!
g"&
6 "
381
8.!
5a"%
2 "
0no
sol
utio
n
9.!
2c&
1"
"5
1310
.!3k
&"
2""
46
11.!
x%
4"
&2
"1
512
.!4x
&4
"&
4 "
05
13.
"4
144
14. #
$"3
27
15.x
"!
x%
2"
216
.d"
!12
&"
d"3
17.!
6x&
9"
"x
318
.!6p
&"
8""
p2,
4
19.!
x%
5"
"x
&1
420
.!8
&c
""
c&
88
21.!
r&
3"
%5
"r
722
.!y
&1
"%
3 "
y5
23.!
5n%
"4"
"n
%2
1,0
24.!
3z&
6"
"z
&2
5,2
m $ 3!
d"$
3
©G
lenc
oe/M
cGra
w-H
ill65
8G
lenc
oe A
lgeb
ra 1
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
.
1.!
&b
""
8"
642.
4!3"
"!
x"48
3.2!
4c"%
3 "
114
4.6
&!
2y""
&2
32
5.!
k%
2"
&3
"7
986.
!m
&5
""
4!3"
53
7.!
6t%
1"
2""
8!6"
628.
!3j
&1
"1"
%2
"9
20
9.!
2x%
1"
5"%
5 "
1877
10. #
$&4
"2
60
11.6#$
&3
"0
12.6
%#$
"&
2no
sol
utio
n
13.y
"!
y%
6"
314
.!15
&"
2x""
x3
15.!
w%
4"
"w
%4
"4,
"3
16.!
17 &
"k"
"k
&5
8
17.!
5m&
"16"
"m
&2
4,5
18.!
24 %
"8q"
"q
%3
"3,
5
19.!
4s%
1"
7"&
s&
3 "
02
20.4
&!
3m%
"28"
"m
"1
21.!
10p
%"
61"&
7 "
p"
6,2
22.!
2x2
&"
9""
x3
ELEC
TRIC
ITY
For
Exe
rcis
es 2
3 an
d 2
4,u
se t
he
foll
owin
g in
form
atio
n.
The
vol
tage
Vin
a c
ircu
it is
giv
en b
y V
"!
PR"
,whe
re P
is t
he p
ower
in w
atts
and
Ris
the
resi
stan
ce in
ohm
s.
23.I
f th
e vo
ltag
e in
a c
ircu
it is
120
vol
ts a
nd t
he c
ircu
it p
rodu
ces
1500
wat
ts o
f po
wer
,wha
tis
the
res
ista
nce
in t
he c
ircu
it?
9.6
ohm
s
24.S
uppo
se a
n el
ectr
icia
n de
sign
s a
circ
uit
wit
h 11
0 vo
lts
and
a re
sist
ance
of
10 o
hms.
How
muc
h po
wer
will
the
cir
cuit
pro
duce
?12
10 w
atts
FREE
FA
LLF
or E
xerc
ises
25
and
26,
use
th
e fo
llow
ing
info
rmat
ion
.A
ssum
ing
no a
ir r
esis
tanc
e,th
e ti
me
tin
sec
onds
tha
t it
tak
es a
n ob
ject
to
fall
hfe
et c
an b
e
dete
rmin
ed b
y th
e eq
uati
on t
".
25.I
f a
skyd
iver
jum
ps f
rom
an
airp
lane
and
fre
e fa
lls f
or 1
0 se
cond
s be
fore
ope
ning
the
para
chut
e,ho
w m
any
feet
doe
s th
e sk
ydiv
er f
all?
1600
ft
26.S
uppo
se a
sec
ond
skyd
iver
jum
ps a
nd f
ree
falls
for
6 s
econ
ds.H
ow m
any
feet
doe
s th
ese
cond
sky
dive
r fa
ll?57
6 ft
!h"
$4
5r $ 61 # 4
3x $ 3
3s $ 5
Pra
ctic
e (A
vera
ge)
Rad
ical
Equ
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-3
11-3
Answers (Lesson 11-3)
© Glencoe/McGraw-Hill A10 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csR
adic
al E
quat
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-3
11-3
©G
lenc
oe/M
cGra
w-H
ill65
9G
lenc
oe A
lgeb
ra 1
Lesson 11-3
Pre-
Act
ivit
yH
ow a
re r
adic
al e
quat
ion
s u
sed
to
fin
d f
ree-
fall
tim
es?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-3 a
t th
e to
p of
pag
e 59
8 in
you
r te
xtbo
ok.
How
can
you
isol
ate
!h"
on o
ne s
ide
of t
he e
quat
ion?
Mul
tiply
eac
h si
de o
f th
e eq
uatio
n by
4.
Read
ing
the
Less
on
1.To
sol
ve a
rad
ical
equ
atio
n,yo
u fi
rst
isol
ate
the
radi
cal o
n on
e si
de o
f th
e eq
uati
on.W
hydo
you
the
n sq
uare
eac
h si
de o
f th
e eq
uati
on?
to e
limin
ate
the
radi
cal
2.a.
Pro
vide
the
rea
son
for
each
ste
p in
the
sol
utio
n of
the
giv
en r
adic
al e
quat
ion.
!5x
&1
"&
4"
x&
3O
rigi
nal e
quat
ion
!5x
&1
""
x%
1A
dd 4
to
each
sid
e.(!
5x&
1"
)2"
(x%
1)2
Squ
are
each
sid
e.5x
&1
"x2
%2x
%1
Sim
plify
.0
"x2
&3x
%2
Sub
trac
t 5x
and
add
1 to
eac
h si
de.
0"
(x&
1)(x
&2)
Fact
or.
x&
1 "
0or
x&
2 "
0Ze
ro P
rodu
ct P
rope
rty
x"
1x
"2
Sol
ve.
b.To
be
sure
tha
t 1
and
2 ar
e th
e co
rrec
t so
luti
ons,
into
whi
ch e
quat
ion
shou
ld y
ousu
bsti
tute
to
chec
k?th
e or
igin
al e
quat
ion
3.a.
How
do
you
dete
rmin
e w
heth
er a
n eq
uati
on h
as a
n ex
tran
eous
sol
utio
n?
Sub
stitu
te t
he s
olut
ion(
s) in
to t
he o
rigi
nal e
quat
ion.
If a
solu
tion
does
not
satis
fy t
he o
rigi
nal e
quat
ion,
then
it is
an
extr
aneo
us s
olut
ion.
b.Is
it n
eces
sary
to
chec
k al
l sol
utio
ns t
o el
imin
ate
extr
aneo
us s
olut
ions
? E
xpla
in.
Yes;
sinc
e yo
u sq
uare
eac
h si
de o
f a
radi
cal e
quat
ion,
and
squa
ring
each
sid
e ca
n so
met
imes
pro
duce
an
extr
aneo
us s
olut
ion,
you
need
to
che
ck a
ll so
lutio
ns.T
he o
nly
way
to
be s
ure
that
a s
olut
ion
is n
otex
tran
eous
is t
o ch
eck
it in
the
ori
gina
l equ
atio
n.
Hel
ping
You
Rem
embe
r
4.H
ow c
an y
ou u
se t
he le
tter
s IS
C t
o re
mem
ber
the
thre
e st
eps
in s
olvi
ng a
rad
ical
equa
tion
?
Sam
ple
answ
er:I
sola
te t
he r
adic
al o
n on
e si
de o
f th
e eq
uatio
n,S
quar
eea
ch s
ide
to e
limin
ate
the
radi
cal,
and
Che
ck fo
r ex
tran
eous
sol
utio
ns.
©G
lenc
oe/M
cGra
w-H
ill66
0G
lenc
oe A
lgeb
ra 1
Spe
cial
Pol
ynom
ial P
rodu
cts
Som
etim
es t
he p
rodu
ct o
f tw
o po
lyno
mia
ls c
an b
e fo
und
read
ily w
ith
the
use
of o
ne o
f th
e sp
ecia
l pro
duct
s of
bin
omia
ls.
For
exam
ple,
you
can
find
the
squ
are
of a
tri
nom
ial b
y re
calli
ng t
hesq
uare
of
a bi
nom
ial.
Fin
d (
x$
y$
z)2 .
(a%
b)2
"a2
%2
#a
#b
%b2
[(x
%y)
%z]
2"
(x%
y)2
%2(
x%
y)z
%z2
"x2
%2x
y%
y2%
2xz
%2y
z%
z2
Fin
d (
3t$
x$
1)(3
t"
x"
1).
(Hin
t:(3
t%
x%
1)(3
t&
x&
1) is
the
pro
duct
of
a su
m 3
t%
(x%
1)an
d a
diff
eren
ce 3
t&
(x%
1).)
(3t
%x
%1)
(3t
&x
&1)
"[3
t%
(x%
1)][
3t&
(x%
1)]
"9t
2&
(x%
1)2
"9t
2&
x2&
2x
&1
Use
a s
pec
ial
pro
du
ct o
f bi
nom
ials
to
fin
d e
ach
pro
du
ct.
1.(x
%y
&z)
22.
(r%
s%
5)2
x2$
2xy
"2x
z$
y2"
2yz
$z2
r2$
2rs
$s2
$10
r$
10s
$25
3.(b
&3
%d)
24.
(k&
m&
2)2
b2
"6b
$2b
d$
9 "
6d$
d2
k2"
2km
$m
2"
4k$
4m$
4
5.(x
%1
%2b
)(x
%1
&2b
)6.
(y&
2 %
x)(y
&2
&x)
x2$
2x$
1 "
4b2
y2"
4y$
4 "
x2
7.(5
%b
&x)
(5 %
b%
x)8.
(j&
5 &
f)(j
%5
%f)
25 $
10b
$b
2"
x2j2
"f2
"10
f"25
9.[(
x%
y) %
(z%
w)]
[(x
%y)
&(z
%w
)]10
.(2a
%1
%3b
&c)
(2a
%1
&3b
%c)
x2$
2xy
$y2
"z2
"2z
w"
w2
4a2
$4a
$1
"9b
2$
6bc
"c2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-3
11-3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Answers (Lesson 11-3)
© Glencoe/McGraw-Hill A11 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
an
d I
nte
rven
tion
The
Pyt
hago
rean
The
orem
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-4
11-4
©G
lenc
oe/M
cGra
w-H
ill66
1G
lenc
oe A
lgeb
ra 1
Lesson 11-4
The
Pyth
agor
ean
Theo
rem
The
sid
e op
posi
te t
he r
ight
ang
le in
a r
ight
tri
angl
e is
calle
d th
e h
ypot
enu
se.T
his
side
is a
lway
s th
e lo
nges
t si
de o
f a r
ight
tri
angl
e.T
he o
ther
two
side
s ar
e ca
lled
the
legs
of t
he t
rian
gle.
To f
ind
the
leng
th o
f an
y si
de o
f a
righ
ttr
iang
le,g
iven
the
leng
ths
of t
he o
ther
tw
o si
des,
you
can
use
the
Pyt
hag
orea
n T
heo
rem
.
Pyt
hago
rean
The
orem
If a
and
bar
e th
e m
easu
res
of th
e le
gs o
f a r
ight
tria
ngle
and
cis
the
mea
sure
of t
he h
ypot
enus
e, th
en c
2!
a2"
b2.
CB
Ab
ac
Fin
d t
he
len
gth
of
the
hyp
oten
use
of
a ri
ght
tria
ngl
e if
a!
5an
d b
!12
.
c2!
a2"
b2P
ytha
gore
an T
heor
em
c2!
52"
122
a!
5 an
d b
!12
c2!
169
Sim
plify
.
c!
!16
9"
Take
the
squa
re r
oot o
f eac
h si
de.
c!
13
The
leng
th o
f th
e hy
pote
nuse
is 1
3.
Fin
d t
he
len
gth
of
a le
gof
a r
igh
t tr
ian
gle
if a
!8
and
c!
10.
c2!
a2"
b2P
ytha
gore
an T
heor
em
102
!82
"b2
a!
8 an
d c
!10
100
!64
"b2
Sim
plify
.
36 !
b2S
ubtr
act 6
4 fr
om e
ach
side
.
b!
#!
36"Ta
ke th
e sq
uare
roo
t of e
ach
side
.
b!
#6
The
leng
th o
f th
e le
g is
6.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
len
gth
of
each
mis
sin
g si
de.
If n
eces
sary
,rou
nd
to
the
nea
rest
hu
nd
red
th.
1.2.
3.
5045
.83
35.3
6
If c
is t
he
mea
sure
of
the
hyp
oten
use
of
a ri
ght
tria
ngl
e,fi
nd
eac
h m
issi
ng
mea
sure
.If
nec
essa
ry,r
oun
d t
o th
e n
eare
st h
un
dre
dth
.
4.a
!10
,b!
12,c
!?
5.a
!9,
b!
12,c
!?
6.a
!12
,b!
?,c
!16
15.6
215
10.5
8
7.a
!?,
b!
6,c
!8
8.a
!?,
b!
!8",
c!
!18"
9.a
!!
5",b
!!
10",c
!?
5.29
3.16
3.87
25
25c
100
110
a
40
30c
©G
lenc
oe/M
cGra
w-H
ill66
2G
lenc
oe A
lgeb
ra 1
Righ
t Tr
iang
les
If a
and
bar
e th
e m
easu
res
of t
he s
hort
er s
ides
of
a tr
iang
le,c
is t
hem
easu
re o
f th
e lo
nges
t si
de,a
nd c
2!
a2"
b2,t
hen
the
tria
ngle
is a
rig
ht t
rian
gle.
Det
erm
ine
wh
eth
er t
he
foll
owin
g si
de
mea
sure
s fo
rm r
igh
t tr
ian
gles
.
a.10
,12,
14Si
nce
the
mea
sure
of
the
long
est
side
is 1
4,le
t c
!14
,a!
10,a
nd b
!12
.
c2!
a2"
b2P
ytha
gore
an T
heor
em
142
!10
2"
122
a!
10, b
!12
, c!
14
196
!10
0 "
144
Mul
tiply
.
196
$24
4A
dd.
Sinc
e c2
$a2
"b2
,the
tri
angl
e is
not
a r
ight
tri
angl
e.
b.7,
24,2
5Si
nce
the
mea
sure
of
the
long
est
side
is 2
5,le
t c
!25
,a!
7,an
d b
!24
.
c2!
a2"
b2P
ytha
gore
an T
heor
em
252
!72
"24
2a
!7,
b!
24, c
!25
625
!49
"57
6M
ultip
ly.
625
!62
5A
dd.
Sinc
e c2
!a2
"b2
,the
tri
angl
e is
a r
ight
tri
angl
e.
Det
erm
ine
wh
eth
er t
he
foll
owin
g si
de
mea
sure
s fo
rm r
igh
t tr
ian
gles
.
1.14
,48,
50ye
s2.
6,8,
10ye
s3.
8,8,
10no
4.90
,120
,150
yes
5.15
,20,
25ye
s6.
4,8,
4!5"
yes
7.2,
2,!
8"ye
s8.
4,4,
!20"
no9.
25,3
0,35
no
10.2
4,36
,48
no11
.18,
80,8
2ye
s12
.150
,200
,25
0ye
s
13.1
00,2
00,3
00no
14.5
00,1
200,
1300
yes
15.7
00,1
000,
1300
no
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
The
Pyt
hago
rean
The
orem
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-4
11-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 11-4)
© Glencoe/McGraw-Hill A12 Glencoe Algebra 1
Skil
ls P
ract
ice
The
Pyt
hago
rean
The
orem
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-4
11-4
©G
lenc
oe/M
cGra
w-H
ill66
3G
lenc
oe A
lgeb
ra 1
Lesson 11-4
Fin
d t
he
len
gth
of
each
mis
sin
g si
de.
If n
eces
sary
,rou
nd
to
the
nea
rest
hu
nd
red
th.
1.2.
3.
7536
30
4.5.
6.
15.7
59.
8570
If c
is t
he
mea
sure
of
the
hyp
oten
use
of
a ri
ght
tria
ngl
e,fi
nd
eac
h m
issi
ng
mea
sure
.If
nec
essa
ry,r
oun
d t
o th
e n
eare
st h
un
dre
dth
.
7.a
"21
,b"
28,c
"?
358.
a"
6,c
"10
,b"
?8
9.a
"15
,b"
36,c
"?
3910
.a"
16,c
"20
,b"
?12
11.a
"5,
b"
12,c
"?
1312
.b"
6,c
"12
,a"
?!
108
"%
10.3
9
13.a
"11
,b"
4,c
"?
!13
7"
%11
.70
14.a
"8,
b"
10,c
"?
!16
4"
%12
.81
15.a
"19
,b"
!39"
,c"
?20
16.a
"!
12",b
"6,
c"
?!
48"%
6.93
17.c
"!
130
",a
"7,
b"
?9
18.a
"!
6",b
"!
19",c
"?
5
Det
erm
ine
wh
eth
er t
he
foll
owin
g si
de
mea
sure
s fo
rm r
igh
t tr
ian
gles
.Ju
stif
y yo
ur
answ
er.
19.7
,24,
25Ye
s;72
$24
2%
252 .
20.1
5,30
,34
No;
152
$30
2(
342 .
21.1
6,28
,32
No;
162
$28
2(
322 .
22.1
8,24
,30
Yes;
182
$24
2%
302 .
23.1
5,36
,39
Yes;
152
$36
2%
392 .
24.5
,7,!
74"Ye
s;52
$72
%( !
74")2 .
25024
0
a4
9c29
33
b
1634
b
1539a
21
72c
©G
lenc
oe/M
cGra
w-H
ill66
4G
lenc
oe A
lgeb
ra 1
Fin
d t
he
len
gth
of
each
mis
sin
g si
de.
If n
eces
sary
,rou
nd
to
the
nea
rest
hu
nd
red
th.
1.2.
3.
6815
.49
11.3
1
If c
is t
he
mea
sure
of
the
hyp
oten
use
of
a ri
ght
tria
ngl
e,fi
nd
eac
h m
issi
ng
mea
sure
.If
nec
essa
ry,r
oun
d t
o th
e n
eare
st h
un
dre
dth
.
4.a
"24
,b"
45,c
"?
515.
a"
28,b
"96
,c"
?10
0
6.b
"48
,c"
52,a
"?
207.
c"
27,a
"18
,b"
?!
405
"%
20.1
2
8.b
"14
,c"
21,a
"?
!24
5"
%15
.65
9.a
"!
20",b
"10
,c"
?!
120
"%
10.9
5
10.a
"!
75",b
"!
6",c
"?
911
.b"
9x,c
"15
x,a
"?
12x
Det
erm
ine
wh
eth
er t
he
foll
owin
g si
de
mea
sure
s fo
rm r
igh
t tr
ian
gles
.Ju
stif
y yo
ur
answ
er.
12.1
1,18
,21
13.2
1,72
,75
No;
112
$18
2(
212 .
Yes;
212
$72
2%
752 .
14.7
,8,1
115
.9,1
0,!
161
"N
o;72
$82
(11
2 .N
o;92
$10
2(
( !16
1"
)2 .
16.9
,2!
10",1
117
.!7",
2!2",
!15"
Yes;
92$
2(!
10")2
%11
2 .Ye
s;( !
7")2
$( 2
!2")
2%
( !15"
)2 .
18.S
TOR
AG
ET
he s
hed
in S
teph
an’s
bac
k ya
rd h
as a
doo
r th
at m
easu
res
6 fe
et h
igh
and
3 fe
et w
ide.
Step
han
wou
ld li
ke t
o st
ore
a sq
uare
the
ater
pro
p th
at is
7 f
eet
on a
sid
e.W
ill it
fit
thr
ough
the
doo
r di
agon
ally
? E
xpla
in.
No;
the
grea
test
leng
th t
hat
will
fit
thr
ough
the
doo
r is
!45"
%6.
71 f
t.
SCR
EEN
SIZ
ESF
or E
xerc
ises
19–
21,u
se t
he
foll
owin
g in
form
atio
n.
The
siz
e of
a t
elev
isio
n is
mea
sure
d by
the
leng
th o
f th
e sc
reen
’s d
iago
nal.
19.I
f a
tele
visi
on s
cree
n m
easu
res
24 in
ches
hig
h an
d 18
inch
es w
ide,
wha
t si
ze t
elev
isio
n is
it?
30-in
.tel
evis
ion
20.D
arla
tol
d T
ri t
hat
she
has
a 35
-inc
h te
levi
sion
.The
hei
ght
of t
he s
cree
n is
21
inch
es.
Wha
t is
its
wid
th?
28 in
.
21.T
ri t
old
Dar
la t
hat
he h
as a
5-i
nch
hand
held
tel
evis
ion
and
that
the
scr
een
mea
sure
s 2
inch
es b
y 3
inch
es.I
s th
is a
rea
sona
ble
mea
sure
for
the
scr
een
size
? E
xpla
in.
No;
ifth
e sc
reen
mea
sure
s 2
in.b
y 3
in.,
then
its
diag
onal
is o
nly
abou
t 3.
61 in
.
124
b19
11
a
60
32cPra
ctic
e (A
vera
ge)
The
Pyt
hago
rean
The
orem
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-4
11-4
Answers (Lesson 11-4)
© Glencoe/McGraw-Hill A13 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csTh
e P
ytha
gore
an T
heor
em
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-4
11-4
©G
lenc
oe/M
cGra
w-H
ill66
5G
lenc
oe A
lgeb
ra 1
Lesson 11-4
Pre-
Act
ivit
yH
ow i
s th
e P
yth
agor
ean
Th
eore
m u
sed
in
rol
ler
coas
ter
des
ign
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-4 a
t th
e to
p of
pag
e 60
5 in
you
r te
xtbo
ok.
The
dia
gram
in t
he in
trod
ucti
on s
how
s a
righ
t tr
iang
le a
nd p
art
of t
hero
ller
coas
ter.
Whi
ch s
ide
of t
he r
ight
tri
angl
e ha
s a
leng
th a
ppro
xim
atel
yeq
ual t
o th
e le
ngth
of
the
firs
t hi
ll of
the
rol
ler
coas
ter?
the
long
est
side
,whi
ch is
the
sid
e op
posi
te t
he r
ight
ang
le
Read
ing
the
Less
onC
omp
lete
eac
h s
ente
nce
.
1.T
he w
ords
leg
and
hypo
tenu
sere
fer
to t
he s
ides
of
a tr
iang
le.
2.In
a r
ight
tri
angl
e,ea
ch o
f the
tw
o si
des
that
form
the
rig
ht a
ngle
is a
of
the
rig
ht t
rian
gle.
3.T
he lo
nges
t si
de o
f a
righ
t tr
iang
le is
cal
led
the
of t
he r
ight
tri
angl
e.
Wri
te a
n e
quat
ion
th
at y
ou c
ould
sol
ve t
o fi
nd
th
e m
issi
ng
sid
e le
ngt
h o
f ea
chri
ght
tria
ngl
e.
4.5.
6.
c2%
92$
102
102
%62
$b2
102
%a2
$92
7.Su
ppos
e yo
u ar
e gi
ven
thre
e po
siti
ve n
umbe
rs.E
xpla
in h
ow y
ou c
an d
ecid
e w
heth
erth
ese
num
bers
are
the
leng
ths
of t
he s
ides
of
a ri
ght
tria
ngle
.
Sam
ple
answ
er:P
ut t
he n
umbe
rs in
ord
er f
rom
leas
t to
gre
ates
t.S
quar
eth
e th
ree
num
bers
.See
whe
ther
the
sum
of
the
squa
res
of t
he t
wo
smal
ler
num
bers
is e
qual
to
the
squa
re o
f th
e la
rges
t nu
mbe
r.If
it is
,th
en t
he n
umbe
rs a
re t
he le
ngth
s of
the
sid
es o
f a
righ
t tr
iang
le.
Hel
ping
You
Rem
embe
r
8.T
hink
of
a w
ord
or p
hras
e th
at y
ou c
an a
ssoc
iate
wit
h th
e P
ytha
gore
an T
heor
em t
o he
lpyo
u re
mem
ber
the
equa
tion
c2
"a2
%b2
.
Sam
ple
answ
er:T
hink
of
the
wor
d ca
b.Th
e le
tter
s of
this
wor
d ar
e th
esa
me
as th
ose
in th
e eq
uatio
n c
2%
a2$
b2 .
9106
109
10
hypo
tenu
se
leg
righ
t
©G
lenc
oe/M
cGra
w-H
ill66
6G
lenc
oe A
lgeb
ra 1
Pyt
hago
rean
Tri
ples
Rec
all t
he P
ytha
gore
an t
heor
em:
In a
rig
ht t
rian
gle,
the
squa
re o
f th
e le
ngth
of
the
hypo
tenu
se
is e
qual
to
the
sum
of
the
squa
res
of t
he le
ngth
s of
the
legs
.
a2%
b2"
c2
Not
e th
at c
is t
he le
ngth
of
the
hyp
oten
use.
The
inte
gers
3,4
,and
5 s
atis
fy t
he
32%
42"
52
Pyt
hago
rean
the
orem
and
can
be
the
9 %
16"
25le
ngth
s of
the
sid
es o
f a
righ
t tr
iang
le.
25"
25
Fur
ther
mor
e,fo
r an
y po
siti
ve in
tege
r n,
For
n"
2:62
%82
"10
2
the
num
bers
3n,
4n,a
nd 5
nsa
tisf
y th
e 36
%64
"10
0P
ytha
gore
an t
heor
em.
100
"10
0
If t
hree
num
bers
sat
isfy
the
Pyt
hago
rean
the
orem
,the
y ar
e ca
lled
a P
yth
agor
ean
tri
ple.
Her
e is
an
easy
way
to
find
oth
er P
ytha
gore
an t
ripl
es.
The
num
bers
a,b
,and
car
e a
Pyt
hago
rean
tri
ple
if a
"m
2&
n2,b
"2m
n,an
d c
"m
2%
n2,w
here
man
d n
are
rela
tive
ly p
rim
e po
siti
ve in
tege
rs a
nd m
(n.
Ch
oose
m%
5 an
d n
%2.
a"
m2
&n2
b"
2mn
c"
m2
%n2
Ch
eck
202
%21
2!
292
"52
&22
"2(
5)(2
)"
52%
2240
0 %
441
!84
1"
25 &
4"
20"
25 %
484
1"
841
"21
"29
Use
th
e fo
llow
ing
valu
es o
f m
and
nto
fin
d P
yth
agor
ean
tri
ple
s.
1.m
"3
and
n"
22.
m"
4 an
d n
"1
3.m
"5
and
n"
3
5,12
,13
8,15
,17
16,3
0,34
4.m
"6
and
n"
55.
m"
10 a
nd n
"7
6.m
"8
and
n"
5
11,6
0,61
51,1
40,1
4939
,80,
89
ACB a
c b
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-4
11-4
Exam
ple
Exam
ple
Answers (Lesson 11-4)
© Glencoe/McGraw-Hill A14 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
The
Dis
tanc
e Fo
rmul
a
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-5
11-5
©G
lenc
oe/M
cGra
w-H
ill66
7G
lenc
oe A
lgeb
ra 1
Lesson 11-5
The
Dis
tanc
e Fo
rmul
aT
he P
ytha
gore
an T
heor
em c
an b
e us
ed t
o de
rive
the
Dis
tan
ceF
orm
ula
show
n be
low
.The
Dis
tanc
e Fo
rmul
a ca
n th
en b
e us
ed t
o fi
nd t
he d
ista
nce
betw
een
any
two
poin
ts in
the
coo
rdin
ate
plan
e.
Dis
tanc
e Fo
rmul
aT
he d
ista
nce
betw
een
any
two
poin
ts w
ith c
oord
inat
es (
x 1, y
1) a
nd (
x 2, y
2) is
giv
en b
y
d"
!(x
2&
"x 1
)2%
"(y
2&
"y 1
)2"
.
Fin
d t
he
dis
tan
ce b
etw
een
th
ep
oin
ts a
t ("
5,2)
an
d (
4,5)
.
d"
!(x
2&
"x 1
)2%
"(y
2&
"y 1
)2"
Dis
tanc
e F
orm
ula
"!
(4 &
("
&5)
)2"
%(5
&"
2)2
"(x
1, y
1) "
(&5,
2),
(x2,
y2)
"(4
, 5)
"!
92%
3"
2 "S
impl
ify.
"!
81 %
"9"
Eva
luat
e sq
uare
s an
d si
mpl
ify.
"!
90"T
he d
ista
nce
is !
90",o
r ab
out
9.49
uni
ts.
Jill
dra
ws
a li
ne
segm
ent
from
poi
nt
(1,4
) on
her
com
pu
ter
scre
en t
o p
oin
t (9
8,49
).H
ow l
ong
is t
he
segm
ent?
d"
!(x
2&
"x 1
)2%
"(y
2&
"y 1
)2"
"!
(98
&"
1)2
%"
(49
&"
4)2
""
!97
2%
"45
2"
"!
9409
"%
202
"5"
"!
11,4
3"
4"T
he s
egm
ent
is a
bout
106
.93
unit
slo
ng.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
oin
ts w
hos
e co
ord
inat
es a
re g
iven
.E
xpre
ss a
nsw
ers
in s
imp
lest
rad
ical
for
m a
nd
as
dec
imal
ap
pro
xim
atio
ns
rou
nd
edto
th
e n
eare
st h
un
dre
dth
if
nec
essa
ry.
1.(1
,5),
(3,1
)2.
(0,0
),(6
,8)
3.(&
2,&
8),(
7,&
3)
2!5";
4.47
10!
106
";1
0.30
4.(6
,&7)
,(&
2,8)
5.(1
,5),
(&8,
4)6.
(3,&
4),(
&4,
&4)
17!
82";9
.06
7
7.(&
1,4)
,(3,
2)8.
(0,0
),(&
3,5)
9.(2
,&6)
,(&
7,1)
2!5";
4.47
!34"
;5.8
3!
130
";1
1.40
10.(
&2,
&5)
,(0,
8)11
.(3,
4),(
0,0)
12.(
3,&
4),(
&4,
&16
)
!17
3"
;13.
155
!19
3"
;13.
89
13.(
1,&
1),(
3,&
2)14
.(&
2,0)
,(&
3,&
9)15
.(&
9,0)
,(&
2,5)
!5";
2.24
!82"
;9.0
6!
74";8
.60
16.(
2,&
7),(
&2,
&2)
17.(
1,&
3),(
&8,
21)
18.(
&3,
&5)
,(1,
&8)
!41"
;6.4
0!
657
";2
5.63
5
©G
lenc
oe/M
cGra
w-H
ill66
8G
lenc
oe A
lgeb
ra 1
Find
Coo
rdin
ates
If y
ou k
now
the
coo
rdin
ates
of
one
poin
t an
d on
ly o
ne c
oord
inat
e of
a
seco
nd p
oint
,you
can
use
the
Dis
tanc
e Fo
rmul
a to
fin
d th
e m
issi
ng c
oord
inat
e of
the
seco
nd p
oint
.
Fin
d t
he
valu
e of
aif
th
e d
ista
nce
bet
wee
n t
he
poi
nts
at
("3,
"2)
and
(a
,"5)
is
5 u
nit
s.
d"
!(x
2&
"x 1
)2%
"(y
2&
"y 1
)2"
Dis
tanc
e F
orm
ula
5"
!(a
&(
"&
3))2
"%
(&5
"&
(&"
2))2
"(x
1, y
1) "
(&3,
&2)
, (x 2
, y2)
"(a
, &5)
, and
d"
5
5"
!(a
%3
")2
%(
"&
3)2
"S
impl
ify.
5"
!a2
%6
"a
%9
"%
9"
Eva
luat
e sq
uare
s.
5"
!a2
%6
"a
%1
"8"
Sim
plify
.
52"
(!a2
%6
"a
%1
"8")
2S
quar
e ea
ch s
ide.
25"
a2%
6a%
18S
impl
ify.
0"
a2%
6a&
7S
ubtr
act 2
5 fr
om e
ach
side
.
0"
(a%
7)(a
&1)
Fac
tor.
a%
7 "
0or
a&
1 "
0Z
ero
Pro
duct
Pro
pert
y
a"
&7
a"
1
The
val
ue o
f ais
&7
or 1
.
Fin
d t
he
pos
sibl
e va
lues
of
aif
th
e p
oin
ts w
ith
th
e gi
ven
coo
rdin
ates
are
th
ein
dic
ated
dis
tan
ce a
par
t.
1.(1
,a),
(3,&
2);d
"!
5"2.
(0,0
),(a
,4);
d"
53.
(2,&
1),(
a,3)
;d"
5
"1
or "
33
or "
3"
1 or
5
4.(1
,&3)
,(a,
21);
d"
255.
(1,a
),(&
2,4)
;d"
36.
(3,&
4),(
&4,
a);d
"!
65""
6 or
84
"8
or 0
7.(a
,&4)
,(&
3,&
2);d
"!
13"8.
(0,3
),(3
,a);
d"
3!2"
9.(a
,3),
(2,&
4);d
"!
74""
6 or
00
or 6
"3
or 7
10.(
&2,
&5)
,(a,
&2)
;d"
511
.(3,
3),(
&1,
a);d
"5
12.(
&1,
&1)
,(4,
a);d
"!
41""
6 or
20
or 6
"5
or 3
13.(
a,5)
,(&
1,2)
;d"
!45"
14.(
4,&
2),(
a,0)
;d"
2!2"
15.(
&2,
1),(
a,&
3);d
"2!
5""
7 or
52
or 6
"4
or 0
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
The
Dis
tanc
e Fo
rmul
a
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-5
11-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 11-5)
© Glencoe/McGraw-Hill A15 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
The
Dis
tanc
e Fo
rmul
a
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-5
11-5
©G
lenc
oe/M
cGra
w-H
ill66
9G
lenc
oe A
lgeb
ra 1
Lesson 11-5
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
oin
ts w
hos
e co
ord
inat
es a
re g
iven
.E
xpre
ss a
nsw
ers
in s
imp
lest
rad
ical
for
m a
nd
as
dec
imal
ap
pro
xim
atio
ns
rou
nd
edto
th
e n
eare
st h
un
dre
dth
if
nec
essa
ry.
1.(9
,7),
(1,1
)10
2.(5
,2),
(8,&
2)5
3.(1
,&3)
,(1,
4)7
4.(7
,2),
(&5,
7)13
5.(&
6,3)
,(10
,3)
166.
(3,3
),(&
2,3)
5
7.(&
1,&
4),(
&6,
0)!
41"%
6.40
8.(&
2,4)
,(5,
8)!
65"%
8.06
9.(&
3,4)
,(&
2,8)
!17"
%4.
1210
.(5,
&6)
,(7,
&9)
!13"
%3.
61
11.(
4,2)
,(8,
6)4!
2"%
5.66
12.(
5,2)
,(3,
10)
2!17"
%8.
25
13.(
12,&
1),(
4,&
11)
2!41"
%12
.81
14.(
&3,
&1)
,(&
11,3
)4!
5"%
8.94
15.(
9,3)
,(6,
&6)
3!10"
%9.
4916
.(0,
&4)
,(8,
4)8!
2"%
11.3
1
Fin
d t
he
pos
sibl
e va
lues
of
aif
th
e p
oin
ts w
ith
th
e gi
ven
coo
rdin
ates
are
th
ein
dic
ated
dis
tan
ce a
par
t.
17.(
&2,
&5)
,(a,
7);d
"13
a%
"7
or 3
18.(
8,&
2),(
5,a)
;d"
3a
%"
2
19.(
4,a)
,(1,
6);d
"5
a%
2 or
10
20.(
a,3)
,(5,
&1)
;d"
5a
%2
or 8
21.(
1,1)
,(a,
1);d
"4
a%
"3
or 5
22.(
2,a)
,(2,
3);d
"10
a%
"7
or 1
3
23.(
a,2)
,(&
3,3)
;d"
!2"
a%
"4
or "
224
.(&
5,3)
,(&
3,a)
;d"
!5"
a%
2 or
4
©G
lenc
oe/M
cGra
w-H
ill67
0G
lenc
oe A
lgeb
ra 1
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
oin
ts w
hos
e co
ord
inat
es a
re g
iven
.E
xpre
ss a
nsw
ers
in s
imp
lest
rad
ical
for
m a
nd
as
dec
imal
ap
pro
xim
atio
ns
rou
nd
edto
th
e n
eare
st h
un
dre
dth
if
nec
essa
ry.
1.(4
,7),
(1,3
)5
2.(0
,9),
(&7,
&2)
!17
0"
%13
.04
3.(4
,&6)
,(3,
&9)
!10"
%3.
164.
(&3,
&8)
,(&
7,2)
2!29"
%10
.77
5.(0
,&4)
,(3,
2)3!
5"%
6.71
6.(&
13,&
9),(
&1,
&5)
4!10"
%12
.65
7.(6
,2),
%4,&
or 2
.50
8.(&
1,7)
, %,6
&%
1.67
9.%2,
&&, %
1,&!
2"%
1.41
10. %
,&1 &,
%2,&
%1.
89
11.(!
3",3)
,(2!
3",5)
!7"
%2.
6512
.(2!
2",&
1),(
3!2",
3)3!
2"%
4.24
Fin
d t
he
pos
sibl
e va
lues
of
aif
th
e p
oin
ts w
ith
th
e gi
ven
coo
rdin
ates
are
th
ein
dic
ated
dis
tan
ce a
par
t.
13.(
4,&
1),(
a,5)
;d"
10a
%"
4 or
12
14.(
2,&
5),(
a,7)
;d"
15a
%"
7 or
11
15.(
6,&
7),(
a,&
4);d
"!
18"a
%3
or 9
16.(
&4,
1),(
a,8)
;d"
!50"
a%
"5
or "
3
17.(
8,&
5),(
a,4)
;d"
!85"
a%
6 or
10
18.(
&9,
7),(
a,5)
;d"
!29"
a%
"14
or
"4
BA
SEB
ALL
For
Exe
rcis
es 1
9–21
,use
th
e fo
llow
ing
info
rmat
ion
.T
hree
pla
yers
are
war
min
g up
for
a b
aseb
all g
ame.
Pla
yer
B
stan
ds 9
feet
to
the
righ
t an
d 18
feet
in fr
ont
of P
laye
r A
.P
laye
r C
sta
nds
8 fe
et t
o th
e le
ft a
nd 1
3 fe
et in
fron
t of
Pla
yer
A.
19.D
raw
a m
odel
of
the
situ
atio
n on
the
coo
rdin
ate
grid
.A
ssum
e th
at P
laye
r A
is lo
cate
d at
(0,
0).
20.T
o th
e ne
ares
t te
nth,
wha
t is
the
dis
tanc
e be
twee
n P
laye
rs A
and
B a
nd b
etw
een
Pla
yers
A a
nd C
?20
.1 f
t;15
.3 f
t
21.W
hat
is t
he d
ista
nce
betw
een
Pla
yers
B a
nd C
?17
.7 f
t
22.M
APS
Mar
ia a
nd J
acks
on li
ve in
adj
acen
t ne
ighb
orho
ods.
If t
hey
supe
rim
pose
aco
ordi
nate
gri
d on
the
map
of
thei
r ne
ighb
orho
ods,
Mar
ia li
ves
at (&
9,1)
and
Jac
kson
lives
at
(5,&
4).I
f ea
ch u
nit
on t
he g
rid
is e
qual
to
appr
oxim
atel
y 0.
132
mile
,how
far
apar
t do
Mar
ia a
nd J
acks
on li
ve?
abou
t 1.
96 m
i
A(0
, 0)
B(9
, 18)
C("
8, 1
3)
x
y
O4
8
16 12 8 4
"8
"4
4!2"
#3
1 $ 32 $ 3
1 $ 21 $ 2
5 # 31 $ 3
5 # 21 $ 2Pra
ctic
e (A
vera
ge)
The
Dis
tanc
e Fo
rmul
a
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-5
11-5
Answers (Lesson 11-5)
© Glencoe/McGraw-Hill A16 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csTh
e D
ista
nce
Form
ula
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-5
11-5
©G
lenc
oe/M
cGra
w-H
ill67
1G
lenc
oe A
lgeb
ra 1
Lesson 11-5
Pre-
Act
ivit
yH
ow c
an t
he
dis
tan
ce b
etw
een
tw
o p
oin
ts b
e d
eter
min
ed?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-5 a
t th
e to
p of
pag
e 61
1 in
you
r te
xtbo
ok.
Wha
t ar
e th
e co
ordi
nate
s of
poi
nts
A,B
,and
C?
A%
(4,3
),B
%("
3,"
2),a
nd C
%(4
,"2)
Read
ing
the
Less
on
1.Su
ppos
e yo
u w
ant
to u
se t
he D
ista
nce
Form
ula
to f
ind
the
dist
ance
bet
wee
n (6
,4)
and
(2,1
).U
se (x
1,y 1
) "(6
,4)
and
(x2,
y 2) "
(2,1
).C
ompl
ete
the
equa
tion
s by
wri
ting
the
co
rrec
t nu
mbe
rs in
the
bla
nks.
a.x 1
"y 1
"x 2
"y 2
"
b.d
"!"
""
"(
&)2
%(
&)2
2.Su
ppos
e yo
u w
ant
to u
se t
he D
ista
nce
Form
ula
to f
ind
the
dist
ance
bet
wee
n (3
,7)
and
(9,&
2).U
se (
x 1,y
1) "
(3,7
) an
d (x
2,y 2
) "(9
,&2)
.Com
plet
e th
e eq
uati
ons
by w
riti
ng t
heco
rrec
t nu
mbe
rs in
the
bla
nks.
a.x 1
"y 1
"x 2
"y 2
"
b.d
"!"
""
"(
&)2
%(
&)2
3.A
cla
ssm
ate
is u
sing
the
Dis
tanc
e Fo
rmul
a to
fin
d th
e di
stan
ce b
etw
een
two
poin
ts.S
he
has
done
eve
ryth
ing
corr
ectl
y so
far
,and
her
equ
atio
n is
d"
!(&
2 &
"5)
2%
"(7
&"
11)2
".
Thi
s eq
uati
on w
ill g
ive
her
the
dist
ance
bet
wee
n w
hat
two
poin
ts?
(5,1
1) a
nd (
"2,
7)
Hel
ping
You
Rem
embe
r
4.So
met
imes
it is
eas
ier
to r
emem
ber
a fo
rmul
a if
you
can
sta
te it
in w
ords
.How
can
you
stat
e th
e D
ista
nce
Form
ula
in e
asy-
to-r
emem
ber
wor
ds?
Sam
ple
answ
er:S
quar
e th
e di
ffer
ence
bet
wee
n th
e x-
coor
dina
tes,
then
add
that
to
the
squa
re o
f th
e di
ffer
ence
bet
wee
n th
e y-
coor
dina
tes.
Take
the
squa
re r
oot
of t
he s
um t
o fin
d th
e di
stan
ce b
etw
een
the
two
poin
ts.
7"
23
9
"2
97
3
41
62
12
46
x
y
O
A
B
©G
lenc
oe/M
cGra
w-H
ill67
2G
lenc
oe A
lgeb
ra 1
A S
pace
-Sav
ing
Met
hod
Tw
o ar
rang
emen
ts f
or c
ooki
es o
n a
32 c
m b
y 40
cm
coo
kie
shee
t ar
e sh
own
at t
he r
ight
.The
coo
kies
hav
e 8-
cm d
iam
eter
saf
ter
they
are
bak
ed.T
he c
ente
rs o
f th
e co
okie
s ar
e on
the
ve
rtic
es o
f sq
uare
s in
the
top
arr
ange
men
t.In
the
oth
er,t
hece
nter
s ar
e on
the
ver
tice
s of
equ
ilate
ral t
rian
gles
.Whi
char
rang
emen
t is
mor
e ec
onom
ical
? T
he t
rian
gle
arra
ngem
ent
is m
ore
econ
omic
al,b
ecau
se it
con
tain
s on
e m
ore
cook
ie.
In t
he s
quar
e ar
rang
emen
t,ro
ws
are
plac
ed e
very
8 c
m.A
t w
hat
inte
rval
s ar
e ro
ws
plac
ed in
the
tri
angl
e ar
rang
emen
t?
Loo
k at
the
rig
ht t
rian
gle
labe
led
a,b,
and
c.A
leg
aof
the
tria
ngle
is t
he r
adiu
s of
a c
ooki
e,or
4 c
m.T
he h
ypot
enus
e c
is
the
sum
of
two
radi
i,or
8 c
m.U
se t
he P
ytha
gore
an t
heor
em t
o fi
nd b
,the
inte
rval
of
the
row
s.
c2"
a2%
b2
82"
42%
b2
64 &
16 "
b2
!48"
"b
4 !3"
"b
b"
4 !3"
'6.
93
The
row
s ar
e pl
aced
app
roxi
mat
ely
ever
y 6.
93 c
m.
Sol
ve e
ach
pro
blem
.
1.Su
ppos
e co
okie
s w
ith
10-c
m d
iam
eter
s ar
e ar
rang
ed in
the
tr
iang
ular
pat
tern
sho
wn
abov
e.W
hat
is t
he in
terv
al b
of t
he r
ows?
8.66
cm
2.F
ind
the
diam
eter
of
a co
okie
if t
he r
ows
are
plac
ed in
the
tr
iang
ular
pat
tern
eve
ry 3
!3"
cm.
6 cm
3.D
escr
ibe
othe
r pr
acti
cal a
pplic
atio
ns in
whi
ch t
his
kind
of
tria
ngul
ar p
atte
rn c
an b
e us
ed t
o ec
onom
ize
on s
pace
.A
nsw
ers
will
var
y.S
ampl
e an
swer
:pac
kagi
ng c
ans
(b
% ?
21 c
ooki
es
ca
(8
cm
20 c
ooki
es
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-5
11-5
Answers (Lesson 11-5)
© Glencoe/McGraw-Hill A17 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Sim
ilar
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-6
11-6
©G
lenc
oe/M
cGra
w-H
ill67
3G
lenc
oe A
lgeb
ra 1
Lesson 11-6
Sim
ilar
Tria
ngle
s!
RS
Tis
sim
ilar
to !
XY
Z.T
he a
ngle
s of
the
tw
o tr
iang
les
have
equ
al m
easu
re.T
hey
are
calle
d co
rres
pon
din
gan
gles
.The
sid
es o
ppos
ite
the
corr
espo
ndin
g an
gles
are
cal
led
corr
esp
ond
ing
sid
es.
If tw
o tr
iang
les
are
sim
ilar,
then
the
!A
BC
!!
DE
FS
imila
r Tr
iang
les
mea
sure
s of
thei
r co
rres
pond
ing
side
sar
e pr
opor
tiona
l and
the
mea
sure
s of
!
!th
eir
corr
espo
ndin
g an
gles
are
equ
al.
AC
" DF
BC
" EF
AB
" DE
A
BD
E
F
C
30!
30!
60!
60!
SZ
X Y
RT
Det
erm
ine
wh
eth
erth
e p
air
of t
rian
gles
is
sim
ilar
.Ju
stif
y yo
ur
answ
er.
Sinc
e co
rres
pond
ing
angl
es d
o no
t ha
veth
e eq
ual m
easu
res,
the
tria
ngle
s ar
eno
t si
mila
r.
50!
75!
45!
89!
R
S
T
X
Y
Z
Det
erm
ine
wh
eth
er t
he
pai
rof
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
The
mea
sure
of "
G!
180°
#(9
0°$
45°)
!45
°.T
he m
easu
re o
f "I
!18
0°#
(45°
$45
°) !
90°.
Sinc
e co
rres
pond
ing
angl
es h
ave
equa
lm
easu
res,
!E
FG
!!
HIJ
.
90!
45!
45!
45!
F
G
H
J
I
E
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
1.2.
3.
Yes;
cor
resp
ondi
ng
No;
cor
resp
ondi
ng
Yes;
cor
resp
ondi
ng
angl
es h
ave
equa
l an
gles
do
not
have
an
gles
hav
e eq
ual
mea
sure
s.eq
ual m
easu
res.
mea
sure
s.
4.5.
6.
Yes;
cor
resp
ondi
ng
Yes;
cor
resp
ondi
ng
No;
cor
resp
ondi
ng
angl
es h
ave
equa
l an
gles
hav
e eq
ual
angl
es d
o no
t ha
ve
mea
sure
s.m
easu
res.
equa
l mea
sure
s.
20!
30!
120!
115!
45!
45!
80!
55!
40!
30!
30!
110!
90!
45!
60!
60!
30!
30!
30!
120!
©G
lenc
oe/M
cGra
w-H
ill67
4G
lenc
oe A
lgeb
ra 1
Find
Unk
now
n M
easu
res
If s
ome
of t
he m
easu
rem
ents
are
kno
wn,
prop
orti
ons
can
be u
sed
to f
ind
the
mea
sure
s of
the
oth
er s
ides
of
sim
ilar
tria
ngle
s.
IND
IREC
T M
EASU
REM
ENT
!A
BC
!!
AE
Din
th
e fi
gure
at
the
righ
t.F
ind
th
e h
eigh
t of
th
e ap
artm
ent
buil
din
g.
Let
BC
!x.
! !E
D!
7, A
D!
25, A
C!
300
25x
!21
00F
ind
the
cros
s pr
oduc
ts.
x!
84T
he a
part
men
t bu
ildin
g is
84
met
ers
high
.
For
eac
h s
et o
f m
easu
res,
fin
d t
he
mea
sure
s of
th
e m
issi
ng
sid
es i
f !
AB
C!
!D
EF
.
1.c
!15
,d!
8,e
!6,
f!
10a
"12
;b"
9
2.c
!20
,a!
12,b
!8,
f!
15d
"9;
e"
6
3.a
!8,
d!
8,e
!6,
f!
7b
"6;
c"
7
4.a
!20
,d!
10,e
!8,
f!
10b
"16
;c"
20
5.c
!5,
d!
10,e
!8,
f!
8a
";b
"5
6.a
!25
,b!
20,c
!15
,f!
12d
"20
;e"
16
7.b
!8,
d!
8,e
!4,
f!
10a
"16
;c"
20
8.IN
DIR
ECT
MEA
SUR
EMEN
TB
ruce
like
s to
am
use
his
brot
her
by s
hini
ng a
fla
shlig
ht o
n hi
s ha
nd a
ndm
akin
g a
shad
ow o
n th
e w
all.
How
far
is it
fro
m
the
flas
hlig
ht t
o th
e w
all?
51.6
in.o
r 4.
3 ft
9.IN
DIR
ECT
MEA
SUR
EMEN
TA
for
est
rang
er u
ses
sim
ilar
tria
ngle
s to
fin
d th
e he
ight
of
a tr
ee.F
ind
the
heig
ht o
f th
e tr
ee.
60 f
t
12 ft
20 ft
x
100
ft
5 in
.6
in.
x in.
4 ft
Note
: Not
dra
wn
to s
cale
25 # 4
A
ca
be
fd
CD
F
EB
25 " 300
7 " x
AD
" AC
ED
" BC
25 m
Note
: Not
dra
wn
to s
cale
275
m
7 m
ADE
CB
x
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sim
ilar T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-6
11-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 11-6)
© Glencoe/McGraw-Hill A18 Glencoe Algebra 1
Skil
ls P
ract
ice
Sim
ilar T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-6
11-6
©G
lenc
oe/M
cGra
w-H
ill67
5G
lenc
oe A
lgeb
ra 1
Lesson 11-6
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
1.2.
Yes;
!A
%!
D%
90°;
!B
%N
o;!
Y%
180°
"(6
0°$
60°)
%60
°.18
0°"
(90°
$40
°) %
50°
%!
E;
Sin
ce "
UV
Wha
s a
57°
angl
e,bu
t !
F%
180°
"(9
0°$
50°)
%40
°%
"X
YZ
does
not
,cor
resp
ondi
ng
!C
.Sin
ce t
he c
orre
spon
ding
an
gles
do
not
all h
ave
equa
l an
gles
hav
e eq
ual m
easu
res,
mea
sure
s,an
d th
e tr
iang
les
are
not
"A
BC
&"
DE
F.si
mila
r.
3.4.
No;
!F
%18
0°"
(45°
$40
°) %
Yes;
!G
%18
0°"
(65°
$52
°) %
63°
%95
°.S
ince
"H
JKha
s a
90°
!K
;!J
%18
0°"
(63°
$52
°) %
65°
%an
gle,
but
"E
FGdo
es n
ot,
!F
;!E
%!
H%
52°.
Sin
ce t
he
corr
espo
ndin
g an
gles
do
not
all
corr
espo
ndin
g an
gles
hav
e eq
ual
have
equ
al m
easu
res,
and
the
m
easu
res,
"E
FG&
"H
JK.
tria
ngle
s ar
e no
t si
mila
r.
For
eac
h s
et o
f m
easu
res
give
n,f
ind
th
e m
easu
res
of t
he
mis
sin
g si
des
if
"P
QR
&"
ST
U.
5.r
"4,
s"
6,t
"3,
u"
2p
%12
,q%
6
6.t
"8,
p"
21,q
"14
,r"
7u
%4,
s%
12
7.p
"15
,q"
10,r
"5,
s"
6t%
4,u
%2
8.p
"48
,s"
16,t
"8,
u"
4r
%12
,q%
24
9.q
"6,
s"
2,t
",u
"r
%2,
p%
8
10.p
"3,
q"
2,r
"1,
u"
s%
1,t%
11.p
"14
,q"
7,u
"2.
5,t
"5
r%
3.5,
s%
10
12.r
"6,
s"
3,t
",u
"p
%8,
q%
79 $ 4
21 $ 8
2 # 31 $ 31 $ 2
3 $ 2
PR
Q
q
rp
SU
T
t
us
52)
52)
63)
65)
FE
HJ
KG
40)
40)
45)
F
EG
H
JK
60)
60)
57)
60)
XV
W
U
Z
Y
40)
50)
A
DF
E C
B
©G
lenc
oe/M
cGra
w-H
ill67
6G
lenc
oe A
lgeb
ra 1
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
1.2.
Yes;
!Q
%!
T%
90°;
!P
%N
o;!
C%
180°
"(4
7°$
80°)
%53
°.18
0°"
(90°
$31
°) %
59°
%!
S;
Sin
ce "
FGH
has
a 56
°an
gle,
but
!U
%18
0°"
(90°
$59
°) %
31°
%"
CD
Edo
es n
ot,c
orre
spon
ding
!
R.S
ince
the
cor
resp
ondi
ng
angl
es d
o no
t al
l hav
e eq
ual
angl
es h
ave
equa
l mea
sure
s,m
easu
res,
and
the
tria
ngle
s ar
e no
t "
PQ
R&
"S
TU.
sim
ilar.
For
eac
h s
et o
f m
easu
res
give
n,f
ind
th
e m
easu
res
of t
he
mis
sin
g si
des
if
"A
BC
&"
DE
F.
3.c
"4,
d"
12,e
"16
,f"
8a
%6,
b%
8
4.e
"20
,a"
24,b
"30
,c"
15d
%16
,f%
10
5.a
"10
,b"
12,c
"6,
d"
4e
%4.
8,f%
2.4
6.a
"4,
d"
6,e
"4,
f"
3c
%2,
b%
7.b
"15
,d"
16,e
"20
,f"
10a
%12
,c%
8.a
"16
,b"
22,c
"12
,f"
8d
%,e
%
9.a
",b
"3,
f"
,e"
7c
%,d
%
10.c
"4,
d"
6,e
"5.
625,
f"
12a
%2,
b%
1.87
5
11.S
HA
DO
WS
Supp
ose
you
are
stan
ding
nea
r a
build
ing
and
you
wan
t to
kno
w it
s he
ight
.T
he b
uild
ing
cast
s a
66-f
oot
shad
ow.Y
ou c
ast
a 3-
foot
sha
dow
.If
you
are
5 fe
et 6
inch
esta
ll,ho
w t
all i
s th
e bu
ildin
g?12
1 ft
12.M
OD
ELS
Tru
ss b
ridg
es u
se t
rian
gles
in t
heir
sup
port
bea
ms.
Mol
ly m
ade
a m
odel
of
atr
uss
brid
ge in
the
sca
le o
f 1
inch
"8
feet
.If
the
heig
ht o
f th
e tr
iang
les
on t
he m
odel
is4.
5 in
ches
,wha
t is
the
hei
ght
of t
he t
rian
gles
on
the
actu
al b
ridg
e?36
ft
35 # 633 # 14
11 $ 25 $ 2
44 # 332 # 3
15 # 2
8 # 3
DF
E
e
fd
AC
B
b
ca
80) 47)
47)
56)
EH
F
GD
C
31)
59)
RQ
STU
P
Pra
ctic
e (A
vera
ge)
Sim
ilar T
rian
gles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-6
11-6
Answers (Lesson 11-6)
© Glencoe/McGraw-Hill A19 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csS
imila
r Tri
angl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-6
11-6
©G
lenc
oe/M
cGra
w-H
ill67
7G
lenc
oe A
lgeb
ra 1
Lesson 11-6
Pre-
Act
ivit
yH
ow a
re s
imil
ar t
rian
gles
rel
ated
to
ph
otog
rap
hy?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-6 a
t th
e to
p of
pag
e 61
6 in
you
r te
xtbo
ok.
How
wou
ld y
ou d
escr
ibe
the
shap
es a
nd s
izes
of
the
figu
res
in t
he d
iagr
am?
Sam
ple
answ
er:T
he f
igur
es h
ave
the
sam
e sh
ape,
but
they
do
not
have
the
sam
e si
ze.
Read
ing
the
Less
onC
omp
lete
eac
h s
ente
nce
.
1.In
sim
ilar
tria
ngle
s,th
e an
gles
of
the
two
tria
ngle
s ca
n be
mat
ched
so
that
angl
es h
ave
equa
l .
2.If
the
ang
les
of o
ne t
rian
gle
do n
ot h
ave
the
sam
e m
easu
res
as t
he a
ngle
s of
a s
econ
d
tria
ngle
,the
n th
e tw
o an
gles
are
not
.
3.If
tw
o tr
iang
les
have
the
sam
e si
ze a
nd s
hape
,the
n th
e m
easu
res
of t
he c
orre
spon
ding
side
s ar
e .
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.E
xpla
in h
ow y
ou k
now
th
atyo
ur
answ
er i
s co
rrec
t.
4.N
ot s
imila
r;th
e an
gles
can
not
be m
atch
ed s
o th
atco
rres
pond
ing
angl
es h
ave
the
sam
e m
easu
res.
5.N
ot s
imila
r;th
e si
des
cann
ot b
e m
atch
ed t
oha
ve c
orre
spon
ding
sid
es p
ropo
rtio
nal.
6.S
imila
r;th
e an
gles
can
be
mat
ched
so
that
corr
espo
ndin
g an
gles
hav
e eq
ual m
easu
res.
Hel
ping
You
Rem
embe
r7.
How
can
you
use
the
idea
tha
t th
e co
rres
pond
ing
side
s of
sim
ilar
tria
ngle
s ar
epr
opor
tion
al t
o he
lp y
ou r
emem
ber
how
to
find
the
unk
now
n le
ngth
s of
the
sid
es o
fsi
mila
r tr
iang
les?
Sam
ple
answ
er:S
ince
the
cor
resp
ondi
ng s
ides
of
sim
ilar
tria
ngle
s ar
e pr
opor
tiona
l,us
e a
prop
ortio
nto
fin
d th
e le
ngth
s of
the
unkn
own
side
s.
35!
35!
20!
125!
125!
20!
4
33.
5
8
67.
5
32!
30!
58!
60!
prop
ortio
nal
sim
ilar
mea
sure
sco
rres
pond
ing
©G
lenc
oe/M
cGra
w-H
ill67
8G
lenc
oe A
lgeb
ra 1
AC
urio
us C
onst
ruct
ion
Man
y m
athe
mat
icia
ns h
ave
been
inte
rest
ed in
w
ays
to c
onst
ruct
the
num
ber
!.H
ere
is o
ne s
uch
geom
etri
c co
nstr
ucti
on.
In t
he d
raw
ing,
tria
ngle
s A
BC
and
AD
Ear
e ri
ght
tria
ngle
s.T
he le
ngth
of A !
D!eq
uals
the
leng
th o
f A!C!
and
F!B!is
par
alle
l to
E!G!
.
The
leng
th o
f B!G!
give
s a
deci
mal
app
roxi
mat
ion
of t
he f
ract
iona
l par
t of
!to
six
dec
imal
pla
ces.
Fol
low
th
e st
eps
to f
ind
th
e le
ngt
h o
f B!
G!.R
oun
dto
sev
en d
ecim
al p
lace
s.
1.U
se t
he le
ngth
of B!
C!an
d th
e P
ytha
gore
an
The
orem
to
find
the
leng
th o
f A !C!
.
AC
""#
##
12#
$%2
"1.
3287
682
2.F
ind
the
leng
th o
f A!D!
.
AD
"A
C"
1.3
2876
82
3.U
se t
he le
ngth
of A!
D!an
d th
e P
ytha
gore
an T
heor
em t
o fi
nd t
he le
ngth
of A!
E!.
AE
""#
##
(AD
)2#
$%2
"1.
4197
271
4.T
he s
ides
of
the
sim
ilar
tria
ngle
s F
ED
and
DE
Aar
e in
pro
port
ion.
So,
".
Fin
d th
e le
ngth
of F !
E!.
FE"
FE"
0.17
6090
2
5.F
ind
the
leng
th o
f A!F!.
AF
"A
E$
FE"
1.24
3636
9
6.T
he s
ides
of
the
sim
ilar
tria
ngle
s A
FB
and
AE
Gar
e in
pro
port
ion.
So,
".
Fin
d th
e le
ngth
of A !
G!.
AG
""
"1.
1415
929
7.N
ow,f
ind
the
leng
th o
f B!G!
.
BG
"A
G2
AB
"A
G2
1 "
0.14
1592
9
8.T
he v
alue
of !
to s
even
dec
imal
pla
ces
is 3
.141
5927
.Com
pare
the
fra
ctio
nal p
art
of !
wit
h th
e le
ngth
of B!
G!.
0.14
1592
9 $
0.14
1592
7 "
0.00
0000
2, a
n er
ror
of le
ss t
han
1 pa
rt in
a
mill
ion
AE
% AF
AB
&A
E%
%A
F
AB
# AG
AF
# AE
1% 4(
AE
)
0.5
# AE
FE
# 0.5
1 % 2
7 % 8
7 – 8
1 – 2
C
E DG
BA
F
1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-6
11-6
Answers (Lesson 11-6)
© Glencoe/McGraw-Hill A20 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Trig
onom
etri
c R
atio
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-7
11-7
©G
lenc
oe/M
cGra
w-H
ill67
9G
lenc
oe A
lgeb
ra 1
Lesson 11-7
Trig
onom
etri
c Ra
tios
For
each
acu
te
angl
e of
a r
ight
tri
angl
e,ce
rtai
n ra
tios
of
side
leng
ths
are
usef
ul.T
hese
rat
ios
are
calle
d tr
igon
omet
ric
rati
os.T
hree
com
mon
rati
os a
re t
he s
ine,
cosi
ne,
and
tan
gen
t,as
def
ined
at
the
righ
t.
a.F
ind
th
e si
ne,
cosi
ne,
and
tan
gen
t of
!R
of "
RS
T.
Rou
nd
to
the
nea
rest
th
ousa
nd
th.
sin
R"
'0.
436
r"
2!19"
, s"
20
cos
R"
"0.
9t"
18, s
"20
tan
R"
'0.
484
r"
2!19"
, t"
18
b.F
ind
th
e m
easu
re o
f !
Rto
th
e n
eare
st d
egre
e.
tan
R'
0.48
4F
rom
par
t a a
bove
Use
TAN
–1on
a c
alcu
lato
r to
fin
d th
e m
easu
re o
f th
e an
gle
who
se t
ange
nt r
atio
is 0
.484
.K
EY
STR
OK
ES:
[TAN
–1] .
484
25.8
2698
212
or a
bout
26°
Fin
d t
he
sin
e,co
sin
e,an
d t
ange
nt
of e
ach
acu
te a
ngl
e.R
oun
d y
our
answ
ers
to t
he
nea
rest
ten
th
ousa
nd
th.
1.2.
!A
:0.4
455,
0.89
11,0
.500
0!
A:0
.923
1,0.
3846
,2.4
000
!B
:0.8
911,
0.44
55,2
.000
0!
B:0
.384
6,0.
9231
,0.4
167
Use
a c
alcu
lato
r to
fin
d t
he
valu
e of
eac
h t
rigo
nom
etri
c ra
tio
to t
he
nea
rest
ten
thou
san
dth
.
3.si
n 45
°0.
7071
4.co
s 47
°0.
6820
5.ta
n 48
°1.
1106
Use
a c
alcu
lato
r to
fin
d t
he
mea
sure
of
each
an
gle
to t
he
nea
rest
deg
ree.
6.si
n A
"0.
7547
49°
7.ta
n C
"2.
3456
67°
8.co
s B
"0.
6947
46°
9.si
n A
"0.
6589
41°
10.t
an C
"1.
9832
63°
11.c
os B
"0.
0136
89°
513 12
CA
B
4.5
cm10
.1 c
m
9 cm
CB
A
ENTE
R2n
d
2!19"
$18
18 $ 202 !19"
$20
1820
2!"19
SR
T
90)
ACB
b
ac
sine
of
!A
sin
A"
$a c$
cosi
ne o
f !
Aco
s A
"$b c$
tang
ent
of !
Ata
n A
"$a b$
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill68
0G
lenc
oe A
lgeb
ra 1
Solv
e Tr
iang
les
Fin
ding
the
mea
sure
s of
all
angl
es a
nd s
ides
in a
rig
ht t
rian
gle
isca
lled
solv
ing
the
tria
ngl
e.Yo
u ca
n us
e tr
igon
omet
ric
rati
os o
r th
e P
ytha
gore
an T
heor
emto
fin
d m
issi
ng s
ide
mea
sure
s.
Fin
d a
ll o
f th
e m
issi
ng
mea
sure
s in
"A
BC
.St
ep 1
Sinc
e th
e su
m o
f th
e m
easu
res
of t
he a
ngle
s of
a t
rian
gle
is 1
80°,
m"
B"
180°
&90
°&
48°
"42
°.
Step
2F
ind
the
valu
e of
x.
sin
48°
"D
efin
ition
of s
ine
0.74
31 '
Eva
luat
e si
n 48
°.
7.4
'x
Mul
tiply
by
10.
Step
3F
ind
the
valu
e of
y.
cos
48°
"D
efin
ition
of c
osin
e
0.66
91 '
Eva
luat
e co
s 48
°.
6.7
'y
Mul
tiply
by
10.
So t
he m
issi
ng m
easu
res
are
42°,
7.4
in.,
and
6.7
in.
Sol
ve e
ach
tri
angl
e.S
tate
th
e si
de
len
gth
s to
th
e n
eare
st t
enth
an
d t
he
angl
em
easu
res
to t
he
nea
rest
deg
ree.
1.2.
3.
m!
A%
56°,
m!
B%
45°,
m!
B%
37°,
m!
B%
34°,
BC
%10
,A
C%
30.1
,A
B%
36.1
AB
%14
.1B
C%
39.9
4.5.
6.
AC
%22
.6,
AC
%10
.1,
AB
%17
,m
!A
%47
°,A
B%
18.1
,m
!A
%28
°,m
!B
%43
°m
!B
%34
°m
!B
%62
°
8
15CB
A56
)
15C
B
A
2433
CB
A
53)
50
A
B C45
)10
CB
A20
30
A
B C
y $ 10y $ 10x $ 10x $ 10
y10
in.
48)
xCA
B
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Trig
onom
etri
c R
atio
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-7
11-7
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 11-7)
© Glencoe/McGraw-Hill A21 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Trig
onom
etri
c R
atio
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-7
11-7
©G
lenc
oe/M
cGra
w-H
ill68
1G
lenc
oe A
lgeb
ra 1
Lesson 11-7
For
eac
h t
rian
gle,
fin
d s
in N
,cos
N,a
nd
tan
Nto
th
e n
eare
st t
en t
hou
san
dth
.
1.2.
3.
sin
N%
0.38
46,
sin
N%
0.6,
sin
N%
0.88
24,
cos
N%
0.92
31,
cos
N%
0.8,
cos
N%
0.47
06,
tan
N%
0.41
67ta
n N
%0.
75ta
n N
%1.
875
Use
a c
alcu
lato
r to
fin
d t
he
valu
e of
eac
h t
rigo
nom
etri
c ra
tio
to t
he
nea
rest
te
n t
hou
san
dth
.
4.si
n 20
°0.
3420
5.co
s 32
°0.
8480
6.ta
n 75
°3.
7321
7.ta
n 17
°0.
3057
8.si
n 38
°0.
6157
9.co
s 55
°0.
5736
Use
a c
alcu
lato
r to
fin
d t
he
mea
sure
of
each
an
gle
to t
he
nea
rest
deg
ree.
10.s
in A
"0.
7624
50°
11.c
os L
"0.
8691
30°
12.t
an H
"1.
3024
52°
13.t
an V
"0.
5876
30°
14.s
in R
"0.
4320
26°
15.c
os S
"0.
3749
68°
For
eac
h t
rian
gle,
fin
d t
he
mea
sure
of
the
ind
icat
ed a
ngl
e to
th
e n
eare
st d
egre
e.
16.
17.
18.
21°
27°
71°
Sol
ve e
ach
rig
ht
tria
ngl
e.S
tate
th
e si
de
len
gth
s to
th
e n
eare
st t
enth
an
d t
he
angl
em
easu
res
to t
he
nea
rest
deg
ree.
19.
20.
21.
!B
%62
°,!
A%
36°,
!A
%37
°,B
C%
9.4
ft,
AB
%21
.0 c
m,
!B
%53
°,A
C%
17.7
ft
BC
%12
.4 c
mA
B%
35 m
21 m
28 m
ACB
17 c
m
54)
ACB
20 ft
28)
CA
B
412
?
5
?
113
8?
8
15
17
NPR
0.8
0.6
1F
N
E
1026 24
FD
N
©G
lenc
oe/M
cGra
w-H
ill68
2G
lenc
oe A
lgeb
ra 1
For
eac
h t
rian
gle,
fin
d s
in T
,cos
T,a
nd
tan
Tto
th
e n
eare
st t
en t
hou
san
dth
.
1.2.
3.
sin
T%
0.32
43,
sin
T%
0.96
,si
n T
%0.
8944
,co
s T
%0.
9459
,co
s T
%0.
28,
cos
T%
0.44
72,
tan
T%
0.34
29ta
n T
%3.
4286
tan
T%
2
Use
a c
alcu
lato
r to
fin
d t
he
valu
e of
eac
h t
rigo
nom
etri
c ra
tio
to t
he
nea
rest
ten
thou
san
dth
.
4.si
n 85
°0.
9962
5.co
s 5°
0.99
626.
tan
32.5
°0.
6371
Use
a c
alcu
lato
r to
fin
d t
he
mea
sure
of
each
an
gle
to t
he
nea
rest
deg
ree.
7.si
n D
"0.
5000
30°
8.co
s Q
"0.
1123
84°
9.ta
n B
"4.
7465
78°
For
eac
h t
rian
gle,
fin
d t
he
mea
sure
of
the
ind
icat
ed a
ngl
e to
th
e n
eare
st d
egre
e.
10.
55°
11.
72°
12.
41°
Sol
ve e
ach
rig
ht
tria
ngl
e.S
tate
th
e si
de
len
gth
s to
th
e n
eare
st t
enth
an
d t
he
angl
em
easu
res
to t
he
nea
rest
deg
ree.
13.
14.
15.
!A
%39
°,!
B%
26°,
!A
%60
°,A
C%
20.2
yd,
AC
%5.
7 cm
,!
B%
30°,
BC
%16
.4 y
dB
C%
11.7
cm
BC
%26
.0 f
t
HIK
ING
For
Exe
rcis
es 1
6 an
d 1
7,u
se t
he
foll
owin
g in
form
atio
n.
The
10-
mile
Low
er W
est
Rim
tra
il in
Mt.
Zion
Nat
iona
l Par
k as
cend
s 26
40 f
eet.
16.W
hat
is t
he a
vera
ge a
ngle
of
elev
atio
n of
the
hik
e fr
om t
he c
anyo
n fl
oor
to t
he t
op o
f th
eca
nyon
? (H
int:
Dra
w a
dia
gram
in w
hich
the
leng
th o
f th
e tr
ail f
orm
s th
e hy
pote
nuse
of
a ri
ght
tria
ngle
.Con
vert
fee
t to
mile
s.)
abou
t 2.
9°
17.W
hat
is t
he h
oriz
onta
l dis
tanc
e co
vere
d on
the
hik
e?ab
out
10 m
i
18.R
AM
P D
ESIG
NA
n en
gine
er d
esig
ned
the
entr
ance
to
a m
useu
m t
o in
clud
e a
whe
elch
air
ram
p th
at is
15
feet
long
and
for
ms
a 6°
angl
e w
ith
a si
dew
alk
in f
ront
of
the
mus
eum
.H
ow h
igh
does
the
ram
p ri
se?
abou
t 1.
6 ft
15 f
30 ft
BC A
13 c
m64
)
A CB
26 y
d
51)
CB
A
1612
?4042
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9?
16
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B25
724
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P
T24
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X
Y
Pra
ctic
e (A
vera
ge)
Trig
onom
etri
c R
atio
s
NA
ME
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__D
ATE
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PE
RIO
D__
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11-7
11-7
Answers (Lesson 11-7)
© Glencoe/McGraw-Hill A22 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csTr
igon
omet
ric
Rat
ios
NA
ME
____
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____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-7
11-7
©G
lenc
oe/M
cGra
w-H
ill68
3G
lenc
oe A
lgeb
ra 1
Lesson 11-7
Pre-
Act
ivit
yH
ow a
re t
rigo
nom
etri
c ra
tios
use
d i
n s
urv
eyin
g?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 11
-7 a
t th
e to
p of
pag
e 62
3 in
you
r te
xtbo
ok.
In w
hich
yea
rs w
ere
the
mea
sure
men
ts o
f th
e he
ight
of
Mt.
Eve
rest
incl
oses
t ag
reem
ent?
1954
and
199
9
Read
ing
the
Less
on
1.C
ompl
ete
each
sen
tenc
e.
a.T
he le
gs o
f tr
iang
le M
NR
are
segm
ents
an
d .
b.T
he h
ypot
enus
e of
tri
angl
e M
NR
is
.
c.T
he le
g op
posi
te "
Nis
,a
nd t
he le
g ad
jace
nt t
o "
Nis
.
d.T
he le
g op
posi
te "
Mis
,a
nd t
he le
g ad
jace
nt t
o "
Mis
.
2.W
rite
Kor
Sto
com
plet
e ea
ch e
quat
ion.
a.ta
n "
b.si
n "
c.si
n "
d.co
s "
e.co
s "
f.ta
n "
3.If
you
look
str
aigh
t in
fro
nt o
f yo
u an
d th
en u
p,th
e tw
o lin
es o
f si
ght
form
an
angl
e of
(dep
ress
ion/
elev
atio
n).I
f yo
u lo
ok s
trai
ght
in f
ront
of
you
and
then
dow
n,th
e tw
o lin
es o
f si
ght
form
an
angl
e of
(d
epre
ssio
n/el
evat
ion)
.
Hel
ping
You
Rem
embe
r
4.W
hat
is o
ne w
ay y
ou c
an r
emem
ber
the
trig
onom
etri
c ra
tios
sin
e,co
sine
,and
tan
gent
?
Sam
ple
answ
er:A
wel
l-kno
wn
mne
mon
ic d
evic
e is
SO
H-C
AH
-TO
A
(sin
%op
posi
te/h
ypot
enus
e,co
s %
adja
cent
/hyp
oten
use,
t an
%op
posi
te/a
djac
ent)
.
depr
essi
on
elev
atio
n
ST
$ KT
KK
T$ K
SK
ST
$ KS
SS
T$ K
SK
KT
$ KS
SK
T$ T
SS
K
S T
"MR
"NR
"NR
"MR
"MN
NR
MR
M
N R
©G
lenc
oe/M
cGra
w-H
ill68
4G
lenc
oe A
lgeb
ra 1
Mod
ern
Art
The
pai
ntin
g be
low
,apt
ly t
itle
d,“R
ight
Tri
angl
es,”
was
pai
nted
by
that
wel
l-kn
own
arti
st T
wo-
loos
e L
a-R
ecta
ngle
.Usi
ng t
he in
form
atio
nbe
low
,fin
d th
e di
men
sion
s of
thi
s m
aste
rpie
ce.(
Hin
t:T
he t
rian
gle
that
incl
udes
"F
is is
osce
les.
)
tan
A"
tan
B"
tan
C"
tan
D"
tan
E"
4ta
n F
"1
tan
G"
1.W
hat
is t
he le
ngth
of
the
pain
ting
?33
ft
2.W
hat
is t
he w
idth
of
the
pain
ting
?36
ft
1 $ 31 $ 2
1 $ 24 $ 5
2 $ 5
6 ft
B
A
G
C
D
F
E
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
11-7
11-7
Answers (Lesson 11-7)