garis singgung lingkaran 1
DESCRIPTION
Garis singgung lingkaran 1TRANSCRIPT
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MATERI
Garis singgung lingkaran
a. Garis singgung persekutuan di
dalam
b. Garis singgung persekutuan di luar
Lingkaran dalam dan luar segitiga
a. Lingkaran dalam segitiga
b. Lingkaran luar segitiga
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GARIS SINGGUNG LINGKARAN
Garis AB merupakan garis singgung lingkaran
pada titik B, sehingga jari-jari OB tegak lurus
terhadap garis singgung AB, maka panjang
OA dapat dihitung dengan teorema
Pythagoras.
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BA
O•
OA2 = OB2 + AB2
AB2 = OA2 - OB2
OB2 = OA2 - AB2
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Garis Singgung Persekutuan dalam
M N
A
B
AB = Garis singgung persekutuan dalam
MN = Garis pusat persekutuan
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AB adalah garis singgung persekutuan dalam.
AB = CN
AB2 = MN2 - ( r1 + r2 )2
M N
A
B
C
r1
r2
r2
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Garis Singgung Persekutuan Luar
M N
A
B
AB = Garis singgung persekutuan luar
MN = Garis pusat persekutuan
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AB adalah garis singgung persekutuan luar.
AB = CN
AB2 = MN2 - ( r1 - r2 )2
M N
A
BCr1
r2
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Lingkaran Dalam segitiga
A
O
DE
F
C
B
ra
c
b
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A
O
DE
F
C
B
ra
c
b
Titik pusat lingkaran dalam adalah titik perpotongan
garis bagi sudut sudut segitiga.
Keliling ∆ ABC = a + b + c = 2s
Jadi, keliling segitiga = 2s atau s = ½ ( a + b + c ).
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Luas segitiga = ½ alas x tinggi , atau
= s(s – a )(s – b)(s – c )
Jika jari-jari lingkaran dalam adalah r, maka :
r = Luas : ½ keliling atau r = L/s
AF = AE = s - a
BF = BD = s - b
CE = CD = s - c
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Lingkaran Luar segitiga
C
A
O
R
B
ba
c
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Titik pusat lingkaran luar segitiga adalah titik
potong garis sumbu sisi-sisi segitiga OA=OB=OC
= jari-jari lingkaran luar.
Jika jari-jari lingkaran luar adalah R, maka :
R = abc : 4L atau , R = abc / 4L
C
A
OR
B
ba
c
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Soal
1Pada gambar di bawah, garis AB merupakan
garis singgung. Panjang OA = 13 cm dan jari-
jari OB = 5 cm. Hitunglah panjang garis
singgung AB.
BA
O•
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Pembahasan
:Perhatikan Δ OAB siku-siku di titik B
AB2 = OA2 - OB2
= 132 - 52
= 169 - 25
= 144
AB = √ 144 = 12 cm.
Jadi, panjang garis singgung AB = 12 cm.
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Soal 2
M N
A
B
Jika : AM = 6 cm , BN = 3 cm dan MN =
15 cm
Tentukan panjang garis singgung AB.
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AB2 = MN2 -( r1 + r2 )2
= 152 - ( 6 + 3 )2
= 225 – 81 = 144
AB = √ 144 = 12 cm
M N
A
B
Pembahasan :
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Soal 3
M
N
A
B
Jika : AM =13 cm , BN = 6 cm dan MN =
25 cm
Tentukan panjang garis singgung AB.
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Pembahasan :
AB2 = MN2 -( r1 - r2 )2
= 252 - ( 13 - 6 )2
= 625 – 49 = 576
AB = √ 576 = 24 cm
M
N
A
B
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Soal 4
Pada gambar di
samping, panjang PQ =
9 cm, QR = 15 cm.
Hitunglah panjang jari-
jari OU.
P Q
R
T
U
S
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Pembahasan :
PQ = 12 cm dan QR = 15 cm
PR2 = QR2 - PQ2
= 152 - 122
= 225 - 144
= 81
PR = 81 = 9 cm
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Pembahasan :
PQ = 12 cm, QR = 15 cm dan PR = 9
cm
Rd = Luas ABC : ½ keliling
= ( ½ x PQ x PR ) : ½ ( PQ + PR +
QS )
= ( ½ x 12 x 9 ) : ½ ( 12 + 9 + 15
)
= 54 : 18
= 3 cm.
Jadi, panjang jari-jarinya adalah 3 cm.
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Cara cepat :
PQ = 12 cm dan QR = 15 cm
PR2 = QR2 - PQ2
= 152 - 122
= 225 - 144
= 81
PR = 81 = 9 cm
Rd = ½ ( PQ + PR – QR )
= ½ ( 12 + 9 – 15 )
= 3 cm.
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Soal 5
Pada gambar di samping, panjang PQ =10 cm, panjang QR = PR = 13 cm.
Hitunglah panjang jari-jari OP.
PQ
R
O •
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Pembahasan :
PQ = 10 cm dan PR = QR = 13 cmRS2 = PR2 - PS2
= 132 - 52
= 169 - 25= 144
PR = 144 = 12 cm
R
PQ
O •
S
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RL = ( abc ) : 4 L
= ( 10 x 13 x 13 ) : ( 4 x ½ x 10 x 12 )
= 1690 : 240 = 7,04 cm
Jadi, jari-jarinya adalah : 7,04 cm.
R
PQ
O •
S
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Soal 6
Pada gambar di
samping,
panjang PQ =8
cm, PR = 15
cm.
Hitunglah
panjang jari-jari
lingkaran luar.
P Q
R
O •
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Pembahasan :
PQ = 8 cm dan PR = 15
cm
QR2 = PQ2 + PR2
= 152 + 82
= 225 + 64
= 289
QR = 289 = 17 cm
P Q
R
O •
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PQ = 8 cm, PR = 15 cm
dan
QR = 17 cm
Rd = ½ QR
= ½ x 17
= 8,5 cm.
Jadi panjang jari-jari
lingkaran adalah 8,5 cm.
P Q
R
O •
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Soal 7
M N
A
B
Jika : AM = 7 cm , BN = 3 cm dan AB = 24
cm
Tentukan jarak kedua pusatnya (MN).
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MN2 = AB2 + ( r1 + r2 )2
= 242 + ( 7 + 3 )2
= 576 + 100 = 676
MN = √ 676 = 26 cm
Jadi, jarak titik pusatnya = 26 cm.
Pembahasan :
M
N
A
B
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Soal 8
M
N
A
B
Jika : AM =4 cm , BN = 2 cm dan MN = 10
cm
Tentukan panjang garis singgung AB.
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Pembahasan :
AB2 = MN2 -( r1 - r2 )2
= 102 - ( 4 - 2 )2
= 100 – 4 = 96
AB = √ 96 = 9,79
Jadi, panjang AB = 9,79 cm.
M
N
A
B
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Soal 9
M
N
A
B
Jika : AM = 7 cm , MN = 26 cm dan AB = 24
cm
Tentukan panjang jari-jari BN.
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MN2 = AB2 + ( r1 + r2 )2
262 = 242 + ( 7 + r )2
676 = 576 + ( 7 + r )2
( 7 + r )2 = 676 – 576 = 100
( 7 + r ) = 100 = 10
7 + r = 10
r = 10 – 7
r = 3
Jadi, jari-jari BN adalah 3 cm.
Pembahasan :
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Soal 10
M
N
A
B
Jika : BN = 2 cm , AB = 12 cm dan MN =
13 cm
Tentukan panjang AM.
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Pembahasan :
( r1 - r2 )2 = MN2 - AB2
( r1 - 2 )2 = 132 - 122
( r1 - 2 )2 = 169 - 144
= 25
( r1 - 2 ) = 25
r1 - 2 = 5
r1 = 5 + 2 = 7
Jadi, panjang jari-jari AM = 7
cm.
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Catatan Khusus
Jika AB garis singgung persekutuan
dalam.
maka : AB2 = MN2 - ( r1 + r2 )2
Jika AB garis singgung persekutuan
luar.
maka : AB2 = MN2 - ( r1 - r2 )2
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PR LKS
HALMAN 25 NO 1-3
KUMPULKAN SELASA
13 MARET 2012
KERJAKAN DI BUKU
LATIHAN
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