A man standing 'a' metres behind and opposite the middle of a football goalobserves that the angle of elevation of the nearer cross-bar is α and that ofthe further crossbar is β. Show that the length of the field is, a(tanα cotβ +1).
Given:
Let AB and CD be the cross bars of the football goal.
Let O be a point, 'a' metres behind and opposite the middle of the football goal.
Let 'l' metres be the length of the field.
Let AB = CD = p m since AB and CD are the cross bars of the football goal.
In right angled ΔBAO,
AB / AO = tan α
AO = p / tan α
In right angled ΔDCO,
CD / CO = tan β
CO = p / tan β
Length of the field = AO + OC
= a + (l - a)
= ( p / tan α) + ( p / tan β)
By replacing p = a tan α we get,
= a (1 + tan α cotβ) m
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