In an equilateral \[\Delta ABC,\] if \[AD\bot BC,\] then which of the following is true? |
A) \[2A{{B}^{2}}=3A{{D}^{2}}\]
B) \[4A{{B}^{2}}=3A{{D}^{2}}\]
C) \[3\,\,A{{B}^{2}}=4A{{D}^{2}}\]
D) \[3A{{B}^{2}}=2A{{D}^{2}}\]
Correct Answer: C
Solution :
\[AB=BC=CA\] |
\[AD\bot BC\]\[\Rightarrow \]\[BD=DC\] |
In \[\Delta ABD,\]\[A{{B}^{2}}=B{{D}^{2}}+A{{D}^{2}}\] |
\[\Rightarrow \]\[A{{B}^{2}}={{\left( \frac{1}{2}AB \right)}^{2}}+A{{D}^{2}}\] |
\[\Rightarrow \]\[A{{B}^{2}}-\frac{1}{4}A{{B}^{2}}=A{{D}^{2}}\]\[\Rightarrow \]\[3A{{B}^{2}}=4A{{D}^{2}}\] |
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