Challenging Geometry...

May 2010
5
0
The line segments PS, OS, NR, and OR divide the rectangle MNOP into a number of smaller regions. Four of these, two triangles and two quadrilaterals, are shaded in the figure(The shaded regions are the polygons with area 9, 35, and x). The areas of the four shaded regions are 9, 35, 6, and x. Determine the value of x.

Thanks very much...
 

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Opalg

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Aug 2007
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The line segments PS, OS, NR, and OR divide the rectangle MNOP into a number of smaller regions. Four of these, two triangles and two quadrilaterals, are shaded in the figure(The shaded regions are the polygons with area 9, 35, and x). The areas of the four shaded regions are 9, 35, 6, and x. Determine the value of x.
Notice that the area of triangle NOR is half the area of the whole rectangle. So is the area of triangle OSP. And so is the area of the whole rectangle with OSP removed, in other words the sum of the areas of triangles NOS and SMP.

Therefore area of NOR = area of NOS + area of SMP. Can you take it from there?