Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it United States

Problem:

Rectangle $R$ with area $20$ and diagonal of length $7$ is translated $2$ units in some direction to form a new rectangle $R'$ . The vertices of $R$ and $R'$ that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

Solution

Solution:

Dissect the hexagon as shown above, so that it consists of a parallelogram and two triangles which are each half the original rectangle. The parallelogram has side lengths $7$ and $2$ , so its maximum possible area is $14$ . As the two triangles combined always have the same area as the original rectangle, $20$ , the answer is $14+20=34$ .

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