O is the centre of the circle of the circle. BC is a diameter of the circle. OD⊥AB (chord). If OD= 4 cm, BD = 5 cm, then CD=
A
13 cm
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B
√71cm
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C
√89cm
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D
None of these
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Solution
The correct option is D None of these Given−BCisadiameterofthecirclewithcentreO.ABisanotherchordofthesamecircle.OD⊥AB.OD=4cm&BD=6cm.CDhasbeenjoined.Tofindout−CD=?WejoinAC.OD⊥AB⟹∠BDO=90o.i.eΔBDOisarightonewithOBashypotenuse.So,byPythagorastheorem,wegetOB=√BD2+OD2=√62+42cm=√52cm.NowOB=OC=cm(radiioftesamecircle).∴BC=OB+OC=(√52+√52)cm=2√52cm.AlsoDisthemidpointofABsinceOD⊥ABandweknowthattheperpendicular,fromthecentreofacircletoanyofitschords,bisectsthelatter.∴AB=2BD=2×6cm=12cm.Again∠BAC=90osinceitisanangleinasemicircle=90o.∴ΔABCisarightonewithBCasashypotenuse.So,byPythagorastheorem,wegetAC=√BC2−AB2=√(√52)2−122cm=8cm.NowwecnsiderΔADC.Here∠CAD=90o(angleinasemicircle=90o).∴ΔADCisarightonewithCDasashypotenuse.So,byPythagorastheorem,wegetCD=√AC2+AD2=√82+62cm=10cm.Noneoftheoptionscomplywiththisresult.Ans−OptionD.