A circle of radius 2cm Touches a circle of radius 10 cm internally. Determine the length of a tangent segment drawn through the center of the larger circle to smaller circle
Answers
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Answer:
Given−
AcircleC
1
ofradius5cmtouchesanotherbiggercircle
C
2
internallyatA.
C
1
passesthroughthecentreMofC
2
.
ThetangentCDtouchesC
1
atP.
CDmeetsC
2
atD.
Tofindout−
CD=?
Solution−
ADisjoined.
CAisthediameterofC
2
sinceNisthecentreofC
2
.
NowAN=2MN=2×5cm=10cm
ButCN=AN(radiiofthesamecircle).
SoCN=10cmandCM=CN+MN=(10+5)cm=15cm.
AlsoCA=2×AN=2×10cm=20cm.
Again∠CPM=90
o
sincetheradius,meetingthepoint
ofcontactofthetangenttoacircle,makes90
o
angle
withthetangent.
∴ΔCPMisarightonewithCMashypotenuse.
∴CP=
CM
2
−PM
2
=
15
2
−5
2
cm=10
2
cm.
Also∠ADC=90
o
sinceitisanangleinthesemicircle.
∴ΔADCisarightonewithCAashypotenuse
(applyingPythagorastheorm).
NowbetweenΔADC&ΔCPMwehave
∠ADC=90
o
=∠CPM,∠MCPcommon.
∴ΔADC&ΔCPMaresimilar.
i.e
CA
CD
=
CM
CP
⟹CD=
CM
CP
×CA=
15
10
2
×20cm
⟹CD=
3
40
2
cm.
Ans−OptionD.