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Question

In the given figure , AB is a side of a regular 6-sided polygon and AC is a side of a regular 8-sided polygon inscribed in a circle with the centre O. Then (i) ∠AOB (ii) ∠ACB (iii) ∠ABC are respectively

A
70o,40o&44.5o.
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B
60o,30o&22.5o.
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C
20o,60o&11.5o.
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D
80o,50o&28.5o.
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Solution

The correct option is B 60o,30o&22.5o.Given−ABisasideofaregularhexagonandACisasideofaregularoctagon.OisthecentreofacircleofwhichAB&ACarechords.OA,OB&BChavebeenjoined.Tofindout−(i)∠AOB=?∠ACB=?∠ABC=?Solution−WejoinOC.ABisasideofaregularhexagonanditisalsoachordofthecircle.i.ethehexagonhasbeeninscribedinthegivencircle.∴Oisthecentreofthehexagonaswellasofthecircle.Weknowthattheanglesubtendedbyasideofaregularpolygonis360ono.ofsides..........(i)Hereno.ofsides=6.∴Anglesubtendedbythesideofofthehexagonis∠AOB=360o6=60o......(i)SimilarlyACisasideofaregularoctagonanditisalsoachordofthecircle.i.etheoctagonhasbeeninscribedinthegivencircle.∴Oisthecentreoftheoctagonaswellasofthecircle.∴by(i),∠AOC=360o8=45o(hereno.ofsides=8).NowACsubtends∠ABCtothecircumferenceand∠AOCtothecentre.Weknowthattheangle,subtendedbyachordofacircleatthecentre,isdoubletheanglesubtendedbythesamechordatthecircumferenceofthecircle..........(a)∴∠ABC=12∠AOC=12×45o=22.5o.........(iii)SimilarlyABsubtends∠ACBtothecircumferenceand∠AOBtothecentre.∴By(a)∠ACB=12∠AOB=12×60o=30o.........(ii)So∠AOB,∠ACB&∠ABCarerespectively60o,30o&22.5o.Ans−OptionB.

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