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Let ABC be a triangle in which the line ...

Let ABC be a triangle in which the line joining the circumecentre and incentre is parallel to base BC of the triangle. Then answer the following questions :
If ODEI is a square where O and I stands for circumcentre and incentre, respectively and D and E are the point of perpendicular from O and I on the base BC, then

A

`(r )/(R )=(3)/(8)`

B

`(r )/(R )=2-sqrt(3)`

C

`(r )/(R )=sqrt(2)-1`

D

`(r )/(R )=(1)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
C
ODEI is a square , hence, OD = OI
Now, `OI = sqrt(R^(2)-2Rr)`
`therefore sqrt(R^(2)-2Rr) = R cos A`
`rArr R^(2)-2Rr = R^(2)cos^(2)A`
or `1-cos^(2)A=(2r)/(R )` Also `cos A = (r )/(R )`
`rArr 1-((r )/(R ))^(2)=(2r)/(R )`
`rArr ((r )/(R ))^(2)+(2r)/(R )-1=0`
`rArr (r )/(R )= sqrt(2)-1`
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