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Area of a Triangle

A lamp post i...

Question

A lamp post is situated at the middle point m of the side ac of a triangular plot ABC with BC=7CM, CA=8M and AB=9M. Lamp posts subtend an $∠ 15^{\circ}$ at the point B.determine the height of the lamp post.

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Solution

To prove: $t a n (\frac{B - C}{2}) = (\frac{b - c}{b + c}) c o t \frac{A}{2}$
Then, a = k sin A, b = k sin B, c = k sin C
$R . H . S = (\frac{b - c}{b + c}) c o t \frac{A}{2}$
$= (\frac{k s i n B - k s i n C}{k s i n B + k s i n C}) c o t \frac{A}{2}$
$= (\frac{s i n B - s i n C}{s i n B + s i n C}) c o t \frac{A}{2}$
$= [\frac{2 s i n (\frac{B - C}{2}) c o s (\frac{B + C}{2})}{2 s i n (\frac{B + C}{2}) c o s (\frac{B - C}{2})}] c o t \frac{A}{2}$
$= t a n (\frac{B - C}{2}) c o t (\frac{B + C}{2}) c o t \frac{A}{2}$
$= t a n (\frac{B - C}{2}) c o t (\frac{π}{2} - \frac{A}{2}) c o t \frac{A}{2}$
$= t a n (\frac{B - C}{2}) t a n \frac{A}{2} c o t \frac{A}{2}$
$= t a n (\frac{B - C}{2})$
= L.H.S

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