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Chitranjan Bhai Dixit
Asked: 2022-11-07T05:47:52+05:30 In:

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)`

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)`
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1 Answer

  1. Correct Answer – 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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