PRMO 2017 Problem 30 | Similarity Problem

Consider the areas of the four triangles obtained by drawing the diagonals $A C$ and $B D$ of a trapezium $A B C D$. 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.

Challenge and Trill of Pre College Mathematics

PRMO 2017 Problem 30

The maximum possible area is 13

Let us denote by $\triangle ABC$ the area of triangle ABC.

Lets the point of intersection of the diagonals of the trapezium be O

Hence $\frac{\triangle ABO}{\triangle CDO}$ = $\frac{{AB}^{2}}{{CD}^{2}}$

$\triangle ADO$=$\triangle BCO$

$\frac{\triangle ABO}{\triangle ADO} $= $\frac{BO}{OD}$=$\frac{AB}{CD}$

If$y$=$\frac{AB}{CD}$ and $\triangle ABO$ = $x$ then

$\triangle AOB$, $\triangle BOC$, $\triangle COD$, $\triangle AOD$ = $x$, $xy$, $x{y}^{2}$, $xy$

Then consider different cases and find out the different values of $x$, $y$ and the corresponding values of the area of the triangle and find the maximum of them.

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