Unusual Polygon
What is the perimeter of this unusually shaped polygon...
Age
14 to 16
Challenge level
Problem
The diagram shows a polygon ABCDEFG, in which $FG =6$ and $GA=AB=BC=CD=DE=EF$.
Also $BDFG$ is a square.
The area of the whole polygon is exactly twice the area of $BDFG$.
Find the length of the perimeter of the polygon.
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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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The area of square BDFG is $6\times 6 = 36$ square units.
So the total area of the three triangles ABG, BCD and DEF is also $36$ square units.
These three triangles are congruent and so each has an area of $12$ square units.
The area of each triangle is $\frac{1}{2}\times base \times \ height$ and the base is $6$ units and hence we have $\frac{1}{2}\times 6 \times \ height = 12$,
so the height is $4$ units.
Let $X$ be the midpoint of BD. Then CX is perpendicular to the base BD (since BCD is an isosceles triangle).
By Pythagoras' Theorem, $BC = \sqrt{3^2+4^2} = 5$ units.
Therefore the perimeter of ABCDEFG is $6\times 5+ 6 = 36$ units.
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