A) \[3\,\,A{{C}^{2}}\]
B) \[4\,\,A{{C}^{2}}\]
C) \[5\,\,A{{C}^{2}}\]
D) \[6\,\,A{{C}^{2}}\]
Correct Answer: A
Solution :
In right angled \[\Delta ABC\] |
In \[\Delta ABN,\] |
\[A{{N}^{2}}=A{{B}^{2}}+B{{N}^{2}}=A{{B}^{2}}+\frac{B{{C}^{2}}}{4}\] ?(i) |
In \[\Delta CBM\,\,C{{M}^{2}}=B{{C}^{2}}+B{{M}^{2}}=B{{C}^{2}}+\frac{A{{B}^{2}}}{4}\] ?(ii) |
From Eqs. (i) and (ii), we get |
\[A{{N}^{2}}+C{{M}^{2}}=A{{B}^{2}}+\frac{A{{B}^{2}}}{4}+B{{C}^{2}}+\frac{B{{C}^{2}}}{4}\]\[=\frac{5\,(A{{B}^{2}}+B{{C}^{2}})}{4}\]\[\Rightarrow \]\[4\,\,(A{{N}^{2}}+C{{M}^{2}})=5A{{C}^{2}}\] |
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