Question

In a triangle ABC, $ C={{90}^{o}} $ . If r is the in radius and R is the circum radius of the triangle, then $ 2\,(r+R) $ is equal to

• A. $ a+c $
• B. $ a+b+c $
• C. $ b+c $
• D. $ a+b $

Answer 1

Answer: (D)

We know that $ \frac{c}{\sin \,C}=2R $ $ (\because \,\,\angle C={{90}^{o}}) $
$ \Rightarrow $ $ c=\,2R $ ..(i) and $ \tan \frac{C}{2}=\frac{r}{s-c} $
$ \Rightarrow $ $ \tan \,\,{{45}^{o}}=\frac{r}{s-c} $
$ \Rightarrow $ $ r=s-c $
$ \Rightarrow $ $ r=\frac{a+b+c}{2}-c $
$ \Rightarrow $ $ 2r=a+b-c $
On adding Eqs. (i) and (ii), we get $ 2(r+R)=a+b $