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Abcde is a Cyclic Pentagon with Centre of Its Circumcircle at Point O Such that Ab = Bc = Cd and Angle Abc = 120° Calculate: (I) ∠Bec (Ii) ∠Bed - ICSE Class 10 - Mathematics

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Question

ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC = 120°

Calculate:
(i) ∠BEC (ii) ∠BED

 

Solution

i) Join OC and OB.
AB = BC = CD and `∠`ABC  =120°
∴ `∠`BCD = `∠`ABC = 120°
OB and OC are the bisectors of `∠`ABC and `∠`BCDrespectively.
∴ `∠`OBC = `∠`BCO = 60°
In ΔBOC,
`∠`BOC = 180°  - (`∠`OBC + `∠`BOC)
⇒ `∠`BOC =  180° - ( 60° + 60°)
⇒ `∠`BOC =  180°  - 120° = 60°
Arc BC subtends `∠`BOCat the centre and `∠`BEC at the remaining part of the circle.
∴ `∠`BEC = `1/2`  `∠`BOC = `1/2 xx 60° = 30°`

ii) In cyclic quadrilateral BCDE,
`∠`BED +  `∠`BCD = 180°
⇒ `∠`BED + 120°  =  180°
∴ `∠` BED =  60°

 

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Solution Abcde is a Cyclic Pentagon with Centre of Its Circumcircle at Point O Such that Ab = Bc = Cd and Angle Abc = 120° Calculate: (I) ∠Bec (Ii) ∠Bed Concept: Cyclic Properties.
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