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Question

O is the centre of the circle of the circle. BC is a diameter of the circle. ODAB (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
GivenBCisadiameterofthecirclewithcentreO.ABisanotherchordofthesamecircle.ODAB.OD=4cm&BD=6cm.CDhasbeenjoined.TofindoutCD=?WejoinAC.ODABBDO=90o.i.eΔBDOisarightonewithOBashypotenuse.So,byPythagorastheorem,wegetOB=BD2+OD2=62+42cm=52cm.NowOB=OC=cm(radiioftesamecircle).BC=OB+OC=(52+52)cm=252cm.AlsoDisthemidpointofABsinceODABandweknowthattheperpendicular,fromthecentreofacircletoanyofitschords,bisectsthelatter.AB=2BD=2×6cm=12cm.AgainBAC=90osinceitisanangleinasemicircle=90o.ΔABCisarightonewithBCasashypotenuse.So,byPythagorastheorem,wegetAC=BC2AB2=(52)2122cm=8cm.NowwecnsiderΔADC.HereCAD=90o(angleinasemicircle=90o).ΔADCisarightonewithCDasashypotenuse.So,byPythagorastheorem,wegetCD=AC2+AD2=82+62cm=10cm.Noneoftheoptionscomplywiththisresult.AnsOptionD.
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