A and B are centres of two circles touching each other externally at M. line AC and line BD .If AD = 6 cm, BC = 9 cm, then
find the lengths of seg AC and seg BD
Answers
Answer:
Step-by-step explanation:
Given−
P&Qarethecetresoftwocirclesofradius
6cm&9cmrespectively,touchingeachotheratM.
ACisatangenttothebiggercircleatCand
BDisatangenttothesmallercircleatD.
Tofindout−
therespectivelengthsofsegAC&segBD=?
Solution−
AM=AD=6cm(radiiofthesamecircle)&
BM=BC=9cm.(radiiofthesamecircle).
Weknowthattheline,joiningthecentresoftwo
circleswhotoucheachother,passesthroughthe
pointofcontactofthecircles.
∴BothAM&BMlieonthesamelineAB.
SoAB=AM+BM=(6+9)cm=15cm.
Againweknowthatifalinetouchesacircleata
pointthentheradiusthroughthatpointis
perpendiculartothetangentatthatpoint.
∴AD⊥BD&BC⊥AC.
SoΔACBisarightonewithABashypotenuse.
∴ApplyingPythagorasTheorem,weget
AC=
AB
2
−BC
2
=
15
2
−9
2
cm=12cm.
SimilarlyΔADBisarightonewithABashypotenuse.
∴ApplyingPythagorasTheorem,weget
BD=
AB
2
−AD
2
=
15
2
−6
2
cm=3
21
cm.
∴segAC=12cmandsegBD=3
21
cm.
Ans−OptionB.