Question

Write True or False and justify your answer in each of the following :

AB is a diameter of a circle and AC is its chord such that $\angle BAC={30}^0$. If the tangent at C intersects AB extended at D, then BC $=$ BD.

• A. True
• B. False
• C. Ambiguous
• D. Data Insufficient

Answer 1

The correct option is A True

$Given-ABisthediameterofacirclemaking{30}^{o}anglewiththchordAC.PDisthetangentatCwhichmeetstheextendedABatD.Tofindout-thestatementBC=BDistrueorfalse.Justification-\angle ACBistheanglesubtendedbythediameterABonthecircumferenceatC.\therefore \angle ACB={90}^o.....\left(i\right)\left(semicircularangle\right)and\angle CAB={30}^o\left(given\right).\therefore \angle ACB+\angle CAB+\angle ABC={180}^o⟹{30}^o+{90}^o+\angle ABC={180}^o⟹\angle ABC={60}^o........\left(ii\right)But\angle CAB+\angle DBC={180}^o\left(linearpair\right)\therefore From\left(ii\right)\angle DBC={120}^o.....\left(a\right)NowPDisthetangentatC\&ACisachorddrawnfromC.\therefore \angle ACP=\angle BAC(correspondingalt.segmentangles)\therefore \angle ACP={60}^o\left(fromii\right).......\left(iii\right)Now\angle BCD={180}^o-(\angle ACP+\angle ACB)\left(sincetheyareonthesameline\&haveverticesatthesamepoint\right)\therefore \angle BCD={180}^o-(\angle BCD={180}^o-({60}^o+{90}^o)={30}^o(fromi\&iii).........(iv)Soin\Delta BDCwehave\angle BDC={180}^o-{120}^o-{30}^o={30}^o(froma\&iv)i.e\Delta BDCisisosceleswithbaseangles\angle BCD\&\angle BDC.\therefore BC=BD.SothestatementBC=BDistrueAns-True$