Two posts are k metre apart and the height of one is double that of the other. If from the mid-point of the line segment joining their feet, an observer finds the angles of elevation of their tops to be complementary, then find the height of the shorter post.
Answers & Comments
join the point A to B and B to D to make Δ
let <ABC = α and <DBE = 90 - α
in ΔABC
tanα = AC/BC = x/(k/2)
tanα = 2x/k ----------------(1)
in ΔDBE
tan(90-α) = DE/BE = 2x/(k/2)
cotα = 4x/k
1/tanα = 4x/k
put the value of tanα from the (1) equation
1/(2x/k) = 4x/k
k/2x = 4x/k
8x² = k²
x² = k²/8
x = k/2√2 meter
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Let the two posts be AB=2h metres and CD=h metres
BE=EC=
2
k
Let ∠DEC=θ. Then, ∠AEB=90
∘
−θ