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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=btan12(α+β).

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Solution

Lets form the figure,


b=BC=ABAC=lsinαlsinβ

a=PQ=AQAP=lcosβlcosα

ab=lcosβlcosαlsinαlsinβ

ab=cosβcosαsinαsinβ

ab=2sin(α+β2)sin(αβ2)2cos(α+β2)sin(αβ2)

ab=tan(α+β2)

a=btan(α+β2)


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