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In the diagram below, each side of convex quadrilateral  \(ABCD\)is trisected.(For example,\(AP = PQ = QB.\) )

The area of convex quadrilateral \(ABCD\) is  \(180 \). Find the area of the shaded hexagon.

 May 18, 2021
 #1
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By similar triangles, the area of the hexagon is 80.

 May 18, 2021
 #2
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Connect  AC

 

AC  is parallel  to  WT  and PS 

 

Area  of  triangles  ADC   and   ABC    =  180

 

Triangle WDT  is similar to triangle ADC

 

Triangle PBS is similar  to triangle ABC

 

Area  of  triangle WDT  =  (2/3)^2  * area  of  triangle ADC =   (4/9) triangle  ADC

 

Area  of  triangle PBS  =  (2/3)^2  *  area  of ytiangle  ABC  =  (4/9) triangle ABC

 

So

 

(4/9) [ ADC] + (4/9) [ ABC ] + shaded area  =  180

 

(4/9)  (  [ ADC ]  +  [ ABC ] )   +  shaded  area  =  180

 

(4/9)  [ 180 ]  +  shaded area  =  180

 

80  + shaded area  =  180

 

shaded  area  =  180  -  80   =   100

 

 

cool cool cool

 May 18, 2021

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