In a triangle , let . If is the inradius and is the circumradius of the triangle , then equals :

We know by sin c rule as Also as (using )

In a triangle , let . If is the inradius and is the circumra... | JEE Main 2019 Properties of Triangle

Q: JEE Main 2019 Properties of Triangle: In a triangle , let . If is the inradius and is the circumradius of the triangle , then equals :...

The correct answer is A. We know by sin c rule as Also as (using )

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Properties of Triangle

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Question

In a triangle $ABC$ , let $\angle C = {\pi \over 2}$ . If $r$ is the inradius and $R$ is the circumradius of the triangle $ABC$ , then $2(r+R)$ equals :

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Solution

We know by sin c rule ${c \over {\sin C}} = 2R \Rightarrow c = 2R\sin C$ $\Rightarrow c = 2R$ $\left( \, \right.$ as $\,\,\,\angle C = {90^ \circ }$ $\left. \, \right)$ Also $\tan {C \over 2} = {r \over {s - c}}$ $\Rightarrow \tan {\pi \over 4} = {r \over {s - c}}$ $\left( \, \right.$ as $\,\,\,\angle C = {90^ \circ }$ $\left. \, \right)$ $\Rightarrow r = s - c = {{a + b - c} \over 2}$ $\Rightarrow 2r + c = a + b$ $\Rightarrow 2r + 2R = a + b$ (using $c=2R$ )

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