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In an equilateral triangle \(ABC\), \(AD\) is an altitude. Then \(4AD^{2}\) is equal to( )
A. \(2BD^2\)
B. \(BC^2\)
C. \(3AB^2\)
D. \(2DC^2\)
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Question

In an equilateral triangle \(ABC\), \(AD\) is an altitude. Then \(4AD^{2}\) is equal to( )Question: In an equilateral triangle ABC, AD is an altitude. Then 4AD^{2} is equal to(   )A. 2BD^2B. BC^2C. 3AB^2D. 2DC^2
A. \(2BD^2\)
B. \(BC^2\)
C. \(3AB^2\)
D. \(2DC^2\)

Answer

C
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Solution

In an equilateral triangle, all sides are equal.
Solution for In an equilateral triangle ABC, AD is an altitude. Then 4AD^{2} is equal to(   )A. 2BD^2B. BC^2C. 3AB^2D. 2DC^2
In \(\Delta ADC\), applying Pythagoras theorem, we get
\(AD^2=\sqrt{AC^2-DC^2}\)
As, \(DC=\frac{BC}{2}\)
\(\Rightarrow\)  \(AD=\sqrt{AC^2-(\frac{BC}{2})^2}\)
\(\Rightarrow\)  \(AD=\sqrt{AC^2-\frac{BC^2}{4}}\)
\(\Rightarrow\)  \(AD=\sqrt{\frac{4\ AC^2-BC^2}{4}}\)
As all sides are equal i.e. \(AB=BC=CA\)
\(\Rightarrow\)  \(AD=\sqrt{\frac{4\ AB^2-AB^2}{4}}\)
\(\Rightarrow\)  \(AD^2=\frac{4\ AB^2-AB^2}{4}\)
\(\Rightarrow\)  \(4AD^2=3\ AB^2\)
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