In a parallelogram , the longer side is twice the shorter side. Let be the quadrilateral formed by the internal bisectors of the angles of . If the area of is 10, find the area of .
Hint 1: Recall that the internal angle bisectors of a parallelogram form a rectangle in the interior.
Hint 2: Use similar triangles and trigonometric relationships in the parallelogram with side lengths and to show the area ratio.
Hint 3: Show that the area of the interior rectangle is exactly of the area of the parallelogram.
Let the parallelogram be with sides and .
The internal angle bisectors of the parallelogram form a rectangle in the interior.
Let's compute the area of in terms of the area of .
The angle bisectors of adjacent angles of a parallelogram are perpendicular, so is indeed a rectangle.
Through geometric analysis and similar triangles, one can find that the ratio of the area of to the area of is exactly .
Since the area of is , the area of the parallelogram is:
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