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Equal Angles Subtend Equal Sides

A and B are t...

Question

A and B are two points on one bank of a straight river, C and D are two points on the other bank. The direction from C to D is the same as from A to B. If AB = a, $∠ C A D = α$ , $∠ D A B = β$ ,
$∠ C B A = γ$ prove that
$C D = \frac{a s i n α s i n γ}{s i n β . s i n (α + β + γ)}$

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Solution

We have marked the angles as given
$∠ A C B = 180^{o} - (α - β) - γ = 180^{o} - (α + β + γ)$
$s i n A C B = s i n (α + β + γ)$ ...(1)
Using sine formula on $Δ A B C$
$\frac{A B}{s i n A C B} = \frac{A C}{s i n γ}$
$∴ A C = \frac{a s i n γ}{s i n (α + β + γ)}$ , by (1) ...(2)
Again from $Δ A C D$ by sine formula, we have
$\frac{C D}{s i n α} = \frac{A C}{s i n β}$
$∴ C D = \frac{s i n α}{s i n β} . \frac{a s i n γ}{s i n (α + β + γ)}$ by (2)
$= \frac{a s i n α s i n γ}{s i n β s i n (α + β + γ)}$

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