Integer Areas
In this drawing, the segment labeled is four times as long as that labeled ; and the segment labeled is five times as long as that labeled .
Each of the four shaded triangles has an area that is an integer value.
What is the smallest possible value of the sum of the shaded areas ?
The answer is 266.
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1 solution
Call the area of the bottom-left triangle A , and that of the top-right triangle B .
From similarity and the given ratio's, we see that the top-left triangle has area 1 6 A and the bottom-right triangle, 2 5 B .
The top two triangles plus the middle quadrilateral together form a triangle, whose area is half of that of the rectangle. The same is true for the bottom two triangles plus the middle quadrilateral. Hence the top two triangles have the same total area as the bottom two triangles: 1 6 A + B = A + 2 5 B . Solving this shows that A B = 8 5 . The smallest solution in integers is therefore A = 8 ; 1 6 A = 1 2 8 ; B = 5 ; 2 5 B = 1 2 5 ; total = 2 6 6 .