The Broken Bamboo Problem
Triangle geometry through broken bamboo
The broken bamboo problem involves the application of the Pythagorean theorem. It first appeared as problem 13 in the Chiu Chang Suan Shu (The Nine Chapters on the Mathematical Art), likely at the time of the Han Dynasty. The accompanying figure first appeared in Yang Hui's 1261 work A Detailed Analysis of the Mathematical Methods in the Nine Chapters.
The broken bamboo problem considers an upright bamboo stalk 10 chi high that is fixed at the base and "broken" partway up its height. While the rooted portion below the "break" is left in place, the upper fragment of stalk is bent over at the break point until it touches the ground 3 chi to the right of the base. The Pythagorean theorem (known as the gou-go relation) is then applied to find the height of the portion of the stalk that remains erect.