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Question Number 21231 by Tinkutara last updated on 16/Sep/17

Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{areas}\:\mathrm{of}\:\mathrm{the}\:\mathrm{four}\:\mathrm{triangles} \\ $$$$\mathrm{obtained}\:\mathrm{by}\:\mathrm{drawing}\:\mathrm{the}\:\mathrm{diagonals}\:{AC} \\ $$$$\mathrm{and}\:{BD}\:\mathrm{of}\:\mathrm{a}\:\mathrm{trapezium}\:{ABCD}.\:\mathrm{The} \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{these}\:\mathrm{areas},\:\mathrm{taken}\:\mathrm{two}\:\mathrm{at} \\ $$$$\mathrm{time},\:\mathrm{are}\:\mathrm{computed}.\:\mathrm{If}\:\mathrm{among}\:\mathrm{the}\:\mathrm{six} \\ $$$$\mathrm{products}\:\mathrm{so}\:\mathrm{obtained},\:\mathrm{two}\:\mathrm{products}\:\mathrm{are} \\ $$$$\mathrm{1296}\:\mathrm{and}\:\mathrm{576},\:\mathrm{determine}\:\mathrm{the}\:\mathrm{square} \\ $$$$\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{area}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{trapezium}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{integer}. \\ $$

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