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The tilted hourglass

In a square, a diagonal and a line segment cross at the center. They delimit a tilted hourglass. What fraction of the square is shaded?

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Solution

The shaded fraction is 1/3. The tilted hourglass consists of two congruent triangles, which follows from rotational symmetry under a 180-degree rotation around the central point (the common vertex of the triangles).

Suppose the square has a side of length 1, so an area of also 1. The base of each triangle is 2/3. The height is 1/2. So the surface of each triangle equals 1/2 x base x height=1/6. Therefore the total shaded area is 1/3, which is also the fraction.

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