Contest problem 5

**Kanwar245** · March 15th 2012, 10:11 AM

The game Hexagon consists of some pieces, each in a shape of a regular hexagon with the area 6. Each piece is divided by lines into four right triangles, six isosceles triangles and one quadrilateral (see the picture). What is the area of the shaded quadrilateral?

**princeps** · March 15th 2012, 10:47 AM

Originally Posted by Kanwar245

The game Hexagon consists of some pieces, each in a shape of a regular hexagon with the area 6. Each piece is divided by lines into four right triangles, six isosceles triangles and one quadrilateral (see the picture). What is the area of the shaded quadrilateral?

$A_H=\frac{3}{2}a^2 \sqrt 3 \Rightarrow 6=\frac{3}{2}a^2 \sqrt 3 \Rightarrow a^2 \sqrt 3=4 \Rightarrow a^2=\frac{4}{\sqrt 3}$

$A_Q=2\left(\frac{a\cdot a \sqrt 3}{8}\right)=\frac{a^2 \sqrt 3}{4}=1$

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