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PRMO 2017 Problem 30 | Similarity Problem

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Try this triangle area Problem based on Similarity from PRMO - 2017.

Similarity Problem - PRMO 2017 Problem 30

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.

Key Concepts

Similar Triangles

Area of Triangles

Finding the maximum area

Suggested Book | Source | Answer

Challenge and Trill of Pre College Mathematics

PRMO 2017 Problem 30

The maximum possible area is 13

Try with Hints

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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