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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 $a$ slides a distance $b$ down the wall making an angle $β$ with the horizontal. Choose the correct option-

a = b tan (\frac{α + β}{2})

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b = a tan \frac{α + β}{2}

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a = b sin \frac{α + β}{2}

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b = a sin \frac{α + β}{2}

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Solution

The correct option is A $a = b tan (\frac{α + β}{2})$
$I n △ O Q S, cos β = \frac{O S}{Q S} \Rightarrow O S = ℓ cos β$

$I n △ O P R, cos \propto= \frac{O R}{P R} \Rightarrow O R = ℓ cos \propto$

$a = O S - O R = ℓ (cos β - cos \propto)$

$I n △ O P R, sin \propto= \frac{O P}{P R} \Rightarrow O P = ℓ sin \propto$

$△ O Q S, sin β = \frac{O Q}{Q S} \Rightarrow O Q = ℓ sin β$

$b = O P - O Q = ℓ (sin \propto - sin β)$

$\frac{a}{b} = \frac{cos β - cos \propto}{sin \propto - sin β} = \frac{2 sin (\frac{\propto + β}{2}) sin (\frac{\propto - β}{2})}{2 cos (\frac{\propto + β}{2}) sin (\frac{\propto - β}{2})}$

$\frac{a}{b} = tan [\frac{\propto + β}{2}]$

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