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Mathematics

Calculating Heights and Distances

A ladder rest...

Question

A ladder rests against a wall at an angle $α$ to the horizontal. Its foot is pulled away from the wall through a distance $a$ , so that it slides a distance $b$ down the wall making an angle $β$ with the horizontal, show that
$a = b tan \frac{1}{2} (α + β)$ .

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Solution

Lets form the figure,

$b = B C = A B - A C = l sin α - l sin β$
$a = P Q = A Q - A P = l cos β - l cos α$
$\Rightarrow \frac{a}{b} = \frac{l cos β - l cos α}{l sin α - l sin β}$
$\Rightarrow \frac{a}{b} = \frac{cos β - cos α}{sin α - sin β}$
$\Rightarrow \frac{a}{b} = \frac{2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})}{2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2})}$
$\Rightarrow \frac{a}{b} = tan (\frac{α + β}{2})$
$\Rightarrow a = b tan (\frac{α + β}{2})$

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