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.
+128407 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
