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Prove that Any Four Vertices of a Regular Pentagon Are Concylic (Lie on the Same Circle) - ICSE Class 10 - Mathematics

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Question

Prove that any four vertices of a regular pentagon are concylic (lie on the same circle)

Solution

ABCDE is a regular pentagon.

∴ `∠` BAF = `∠`ABC = `∠`BCD = `∠`CDE = `∠`DEA = `((5-2)/5) xx 180° = 180°`

In ΔAED,
AE = ED (Sides of regular pentagon ABCDE)
∴ `∠` EAD =  `∠`EDA

In ΔAED,
 `∠` AED + `∠` EAD + `∠` EDA = 180º
⇒  108º +  `∠` EAD +  `∠` EAD = 180º
⇒ 2 `∠` EAD = 180º −108º = 72º
⇒  `∠` EAD = 36º
∴  `∠` EDA = 36º
 `∠` BAD =  `∠` BAE − `∠` EAD = 108º − 36º = 72º
In quadrilateral ABCD,
`∠`BAD + `∠` BCD = 108º + 72º = 180º
∴ ABCD is a cyclic quadrilateral

 

 

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Solution Prove that Any Four Vertices of a Regular Pentagon Are Concylic (Lie on the Same Circle) Concept: Cyclic Properties.
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