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Two posts are k metres apart and the height of one is double that of the other. If from the middle point of the line joining their feet, an observer finds

the angular elevations of their tops to be complementary, then find the height (in metres) of the shorter post.

|D | | | | A| | 2x | | | | x | | | | C |................k/2................B................k/2...............| E <---------------------------------k--------------------------> 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