A) \[a+b+c\]
B) \[\frac{\sqrt{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}}{2}\]
C) \[\frac{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}{2}\]
D) \[\frac{\sqrt{ab}}{c}\]
Correct Answer: C
Solution :
[c] From figure In \[\Delta ONA,\]\[O{{N}^{2}}={{r}^{2}}-{{a}^{2}}\]and In \[\Delta ONP,\]\[N{{P}^{2}}=O{{P}^{2}}-O{{N}^{2}}\] \[={{c}^{2}}+{{a}^{2}}-{{r}^{2}}-{{r}^{2}}-{{b}^{2}}\] Now, In \[\Delta OMD,\] \[O{{M}^{2}}=N{{P}^{2}}={{r}^{2}}-{{b}^{2}}\] \[\Rightarrow \] \[{{c}^{2}}+{{a}^{2}}-{{r}^{2}}={{r}^{2}}-{{b}^{2}}\] \[\Rightarrow \] \[2{{r}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] \[\Rightarrow \] \[{{r}^{2}}=\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{2}\] \[\Rightarrow \] \[r=\frac{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}{2}\] |
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