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  4. Let ABC be an acute-angled triangle and let D be the mid-point of BC. If AB = AD, then tan(B ) / tan (C) equals

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Q. Let $ABC$ be an acute-angled triangle and let $D$ be the mid-point of $BC$. If $AB = AD$, then $\tan(B ) / \tan (C)$ equals

KVPYKVPY 2013

Solution:

Given, in $\Delta ABC, D$ is mid-point of $BC$ and $AB = AD$
$\therefore \angle B = \angle ADB$
$\theta = \pi - \angle ADB = \pi - B$
$BD = DC = 1 : 1$
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Apply $(m-n) $ theorem,
$( m + n ) \cot \,\theta = n\, \cot\,B - m\,\cot \,C$
$\Rightarrow (1 + 1) \cot \,(\pi - B ) = \cot \,B - \cot \,C$
$[\because m = n = 1]$
$\Rightarrow -2 \,\cot \,B = \cot \,B - \cot\,C$
$[\because \cot \,(\pi - B) = - \cot\,B]$
$\Rightarrow 3\,\cot\,B = \cot\,C$
$\Rightarrow \frac{\tan\,B}{\tan\,C} = 3$