| If \[a+b+c=2s,\] then the value of \[{{(s-a)}^{2}}+{{(s-b)}^{2}}+{{(s-c)}^{2}}\] will be |
A) \[{{s}^{2}}+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
B) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-{{s}^{2}}\]
C) \[{{s}^{2}}-{{a}^{2}}-{{b}^{2}}-{{c}^{2}}\]
D) \[4{{s}^{2}}-{{a}^{2}}-{{b}^{2}}-{{c}^{2}}\]
Correct Answer: B
Solution :
| \[a+b+c=2\,s\]\[\Rightarrow \]\[s=\frac{a+b+c}{2}\] |
| By expanding the expression, |
| \[{{(s-a)}^{2}}+{{(s-b)}^{2}}+{{(s-c)}^{2}}\] |
| \[={{s}^{2}}+{{a}^{2}}-2as+{{s}^{2}}+{{b}^{2}}-2bs+{{s}^{2}}+{{c}^{2}}-2cs\] |
| \[=3{{s}^{2}}+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-2s\,\,(a+b+c)\] |
| \[=3{{s}^{2}}+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-2s\,(2s)\] |
| \[=3{{s}^{2}}+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-4{{s}^{2}}\] |
| \[={{a}^{2}}+{{b}^{2}}+{{c}^{2}}-{{s}^{2}}\] |
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