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Let ABCD be a convex quadrilateral. Let P and Q be points on side \(\overline{AB}\) such that AP = PQ = QB. Similarly, BR = RS = SC, CT = TU = UD, and DV = VW = WA.

The area of quadrilateral ABCD is 180. Find the area of hexagon AWTCSP.

 May 13, 2020
 #1
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The area of hexagon AWTCSP is 92.

 May 13, 2020
 #2
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Draw line segment AC, dividing the figure into two triangles.

In the top triangle, triangle(BPS) ~ triangle(BAC)   [side-angle-side for similarity]

Since BP = (2/3)rds BA, the area of triangle(BPS)  =  (2/3 · 2/3)·triangle(BAC) =  4/9·triangle(BAC)

 

Similarly, the area of triangle(WDT) = 4/9·triangle(ADC).

 

Adding these two parts together, we get the fact that the unshaded area is 4/9ths the quadrilateral.

However, we want the shaded area, so it will be 5/9ths the quadrilateral.

(5/9)·180  =  100

 May 14, 2020

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