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The Door and the Rod

Given a door and a measuring rod of unknown dimensions, the rod is used to measure the door.  It is found that the rod is 4 feet longer than the width of the door, two feet longer than the height and the same length as the diagonal.  What are the dimensions of the door?

Jiuzhang Suanshu (The Nine Chapters on the Mathematical Art), c. 100 BCE

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"The Door and the Rod," Convergence (July 2006)