### Brain Teasers

# Big Polygon

The four edges of a quadrangle have lengths 1, 5, 5, and 7, and the two edges with length 5 are perpendicular. Four squares are built from the four edges. The adjacent vertices of these squares are connected so that a big polygon is formed. Find the area of this big polygon.

### Hint

Since 5^2+5^2=1^2+7^2, by connecting the diagonal of the original quadrangle, we see that the edge with length 1 is also perpendicular to the edge with length 7.### Answer

The big polygon is made up of the original quadrangle, four squares, two right-angled triangles, and two remaining triangles.The area of the original quadrangle is (5*5+1*7)/2=16, which is also the sum of the areas of the two right-angled triangles. The four squares total 1^2+5^2+5^2+7^2=100. The sum of the areas of the two remaining triangles also equal the area of the original quadrangle. This is easily seen by rotating the triangles by 90 degrees, and connecting the other diagonal of the original quadrangle.

Therefore, the total area is 100+16*3=148.

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