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Question

If in a triangle ABC, the line joining the circumcentre O and the incentre I is parallel to BC, then


A

r=RcosA

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B

r=RsinA

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C

R=rcosA

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D

R=rsinA

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Solution

The correct option is A

r=RcosA


Explanation for the correct option.

Find the correct relation.

The given condition that the line joining the circumcentre O and the incentre I is parallel to BC can be shown in the figure as:

As OI is parallel to BC, so O and I are equidistant from BC and thus OL=IM=r(OL⊥BC,IM⊥BC)

Now, as O is the circumcenter so ∠BOC=2A and also

∠BOL=12∠BOC=12×2A=A

Again as OB is the orthoradius, so OB=R.

Now, in â–³OBL,

cosA=OLOB⇒cosA=rR⇒RcosA=r

Hence, the correct option is A.


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